Average Error: 29.2 → 0.8
Time: 17.0s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\mathsf{fma}\left(x, x, \mathsf{fma}\left({x}^{4}, \frac{1}{12}, {x}^{6} \cdot \frac{1}{360}\right)\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
\mathsf{fma}\left(x, x, \mathsf{fma}\left({x}^{4}, \frac{1}{12}, {x}^{6} \cdot \frac{1}{360}\right)\right)
double f(double x) {
        double r3744629 = x;
        double r3744630 = exp(r3744629);
        double r3744631 = 2.0;
        double r3744632 = r3744630 - r3744631;
        double r3744633 = -r3744629;
        double r3744634 = exp(r3744633);
        double r3744635 = r3744632 + r3744634;
        return r3744635;
}
↓
double f(double x) {
        double r3744636 = x;
        double r3744637 = 4.0;
        double r3744638 = pow(r3744636, r3744637);
        double r3744639 = 0.08333333333333333;
        double r3744640 = 6.0;
        double r3744641 = pow(r3744636, r3744640);
        double r3744642 = 0.002777777777777778;
        double r3744643 = r3744641 * r3744642;
        double r3744644 = fma(r3744638, r3744639, r3744643);
        double r3744645 = fma(r3744636, r3744636, r3744644);
        return r3744645;
}

Error

Bits error versus x

Target

Original29.2
Target0.0
Herbie0.8
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.2

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  4. Taylor expanded around 0 0.8

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

  5. Simplified0.8

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

  6. Final simplification0.8

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

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

  :herbie-target
  (* 4 (pow (sinh (/ x 2)) 2))

  (+ (- (exp x) 2) (exp (- x))))