Average Error: 29.4 → 0.6
Time: 1.1m
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\mathsf{fma}\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right), \left(x \cdot x\right), \left(\sqrt{\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \frac{1}{12}, \left(x \cdot x\right)\right)} \cdot \sqrt{\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \frac{1}{12}, \left(x \cdot x\right)\right)}\right)\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
\mathsf{fma}\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right), \left(x \cdot x\right), \left(\sqrt{\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \frac{1}{12}, \left(x \cdot x\right)\right)} \cdot \sqrt{\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \frac{1}{12}, \left(x \cdot x\right)\right)}\right)\right)
double f(double x) {
        double r15700218 = x;
        double r15700219 = exp(r15700218);
        double r15700220 = 2.0;
        double r15700221 = r15700219 - r15700220;
        double r15700222 = -r15700218;
        double r15700223 = exp(r15700222);
        double r15700224 = r15700221 + r15700223;
        return r15700224;
}
↓
double f(double x) {
        double r15700225 = x;
        double r15700226 = r15700225 * r15700225;
        double r15700227 = r15700226 * r15700226;
        double r15700228 = 0.002777777777777778;
        double r15700229 = r15700227 * r15700228;
        double r15700230 = 0.08333333333333333;
        double r15700231 = fma(r15700227, r15700230, r15700226);
        double r15700232 = sqrt(r15700231);
        double r15700233 = r15700232 * r15700232;
        double r15700234 = fma(r15700229, r15700226, r15700233);
        return r15700234;
}

Error

Bits error versus x

Target

Original29.4
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.4

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  5. Applied add-sqr-sqrt0.6

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

  6. Final simplification0.6

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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