Average Error: 29.5 → 0.6
Time: 16.9s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\mathsf{fma}\left({x}^{6}, \frac{1}{360}, \sqrt{\mathsf{fma}\left({x}^{4}, \frac{1}{12}, {x}^{2}\right)} \cdot \sqrt{\mathsf{fma}\left({x}^{4}, \frac{1}{12}, {x}^{2}\right)}\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
\mathsf{fma}\left({x}^{6}, \frac{1}{360}, \sqrt{\mathsf{fma}\left({x}^{4}, \frac{1}{12}, {x}^{2}\right)} \cdot \sqrt{\mathsf{fma}\left({x}^{4}, \frac{1}{12}, {x}^{2}\right)}\right)
double f(double x) {
        double r80802 = x;
        double r80803 = exp(r80802);
        double r80804 = 2.0;
        double r80805 = r80803 - r80804;
        double r80806 = -r80802;
        double r80807 = exp(r80806);
        double r80808 = r80805 + r80807;
        return r80808;
}
↓
double f(double x) {
        double r80809 = x;
        double r80810 = 6.0;
        double r80811 = pow(r80809, r80810);
        double r80812 = 0.002777777777777778;
        double r80813 = 4.0;
        double r80814 = pow(r80809, r80813);
        double r80815 = 0.08333333333333333;
        double r80816 = 2.0;
        double r80817 = pow(r80809, r80816);
        double r80818 = fma(r80814, r80815, r80817);
        double r80819 = sqrt(r80818);
        double r80820 = r80819 * r80819;
        double r80821 = fma(r80811, r80812, r80820);
        return r80821;
}

Error

Bits error versus x

Target

Original29.5
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.5

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

  2. Simplified29.5

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

  3. Taylor expanded around 0 0.6

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

  4. Simplified0.6

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

  6. Applied add-sqr-sqrt0.6

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

  7. Simplified0.6

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

  8. Simplified0.6

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

  9. Final simplification0.6

    \[\leadsto \mathsf{fma}\left({x}^{6}, \frac{1}{360}, \sqrt{\mathsf{fma}\left({x}^{4}, \frac{1}{12}, {x}^{2}\right)} \cdot \sqrt{\mathsf{fma}\left({x}^{4}, \frac{1}{12}, {x}^{2}\right)}\right)\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

  :herbie-target
  (* 4.0 (pow (sinh (/ x 2.0)) 2.0))

  (+ (- (exp x) 2.0) (exp (- x))))