Average Error: 30.2 → 0.7
Time: 5.2s
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 r134499 = x;
        double r134500 = exp(r134499);
        double r134501 = 2.0;
        double r134502 = r134500 - r134501;
        double r134503 = -r134499;
        double r134504 = exp(r134503);
        double r134505 = r134502 + r134504;
        return r134505;
}
↓
double f(double x) {
        double r134506 = x;
        double r134507 = 0.002777777777777778;
        double r134508 = 6.0;
        double r134509 = pow(r134506, r134508);
        double r134510 = 0.08333333333333333;
        double r134511 = 4.0;
        double r134512 = pow(r134506, r134511);
        double r134513 = r134510 * r134512;
        double r134514 = fma(r134507, r134509, r134513);
        double r134515 = fma(r134506, r134506, r134514);
        return r134515;
}

Error

Bits error versus x

Target

Original30.2
Target0.0
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 30.2

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

  2. Taylor expanded around 0 0.7

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

  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 2019353 +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))))