Average Error: 30.0 → 0.7
Time: 4.5s
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 r143562 = x;
        double r143563 = exp(r143562);
        double r143564 = 2.0;
        double r143565 = r143563 - r143564;
        double r143566 = -r143562;
        double r143567 = exp(r143566);
        double r143568 = r143565 + r143567;
        return r143568;
}
↓
double f(double x) {
        double r143569 = x;
        double r143570 = 0.002777777777777778;
        double r143571 = 6.0;
        double r143572 = pow(r143569, r143571);
        double r143573 = 0.08333333333333333;
        double r143574 = 4.0;
        double r143575 = pow(r143569, r143574);
        double r143576 = r143573 * r143575;
        double r143577 = fma(r143570, r143572, r143576);
        double r143578 = fma(r143569, r143569, r143577);
        return r143578;
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 30.0

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

  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 2020057 +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))))