Average Error: 29.4 → 0.7
Time: 27.1s
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 r3569298 = x;
        double r3569299 = exp(r3569298);
        double r3569300 = 2.0;
        double r3569301 = r3569299 - r3569300;
        double r3569302 = -r3569298;
        double r3569303 = exp(r3569302);
        double r3569304 = r3569301 + r3569303;
        return r3569304;
}
↓
double f(double x) {
        double r3569305 = x;
        double r3569306 = r3569305 * r3569305;
        double r3569307 = r3569306 * r3569306;
        double r3569308 = r3569307 * r3569306;
        double r3569309 = 0.002777777777777778;
        double r3569310 = 0.08333333333333333;
        double r3569311 = fma(r3569307, r3569310, r3569306);
        double r3569312 = fma(r3569308, r3569309, r3569311);
        return r3569312;
}

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 2019168 +o rules:numerics
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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