Average Error: 29.4 → 0.7
Time: 23.6s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right), \frac{1}{360}, \mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right), \frac{1}{12}, x \cdot x\right)\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right), \frac{1}{360}, \mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right), \frac{1}{12}, x \cdot x\right)\right)
double f(double x) {
        double r4612567 = x;
        double r4612568 = exp(r4612567);
        double r4612569 = 2.0;
        double r4612570 = r4612568 - r4612569;
        double r4612571 = -r4612567;
        double r4612572 = exp(r4612571);
        double r4612573 = r4612570 + r4612572;
        return r4612573;
}
↓
double f(double x) {
        double r4612574 = x;
        double r4612575 = r4612574 * r4612574;
        double r4612576 = r4612575 * r4612575;
        double r4612577 = r4612576 * r4612575;
        double r4612578 = 0.002777777777777778;
        double r4612579 = 0.08333333333333333;
        double r4612580 = fma(r4612576, r4612579, r4612575);
        double r4612581 = fma(r4612577, r4612578, r4612580);
        return r4612581;
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 29.4

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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