Average Error: 30.1 → 0.7
Time: 15.2s
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 r76511 = x;
        double r76512 = exp(r76511);
        double r76513 = 2.0;
        double r76514 = r76512 - r76513;
        double r76515 = -r76511;
        double r76516 = exp(r76515);
        double r76517 = r76514 + r76516;
        return r76517;
}
↓
double f(double x) {
        double r76518 = x;
        double r76519 = 0.002777777777777778;
        double r76520 = 6.0;
        double r76521 = pow(r76518, r76520);
        double r76522 = 0.08333333333333333;
        double r76523 = 4.0;
        double r76524 = pow(r76518, r76523);
        double r76525 = r76522 * r76524;
        double r76526 = fma(r76519, r76521, r76525);
        double r76527 = fma(r76518, r76518, r76526);
        return r76527;
}

Error

Bits error versus x

Target

Original30.1
Target0.0
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 30.1

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

  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 2019303 +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))))