Average Error: 29.3 → 0.7
Time: 4.4s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\mathsf{fma}\left(x, x, \mathsf{fma}\left(\frac{1}{360}, {x}^{6}, \frac{1}{12} \cdot {x}^{4}\right)\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
\mathsf{fma}\left(x, x, \mathsf{fma}\left(\frac{1}{360}, {x}^{6}, \frac{1}{12} \cdot {x}^{4}\right)\right)
double f(double x) {
        double r97998 = x;
        double r97999 = exp(r97998);
        double r98000 = 2.0;
        double r98001 = r97999 - r98000;
        double r98002 = -r97998;
        double r98003 = exp(r98002);
        double r98004 = r98001 + r98003;
        return r98004;
}
↓
double f(double x) {
        double r98005 = x;
        double r98006 = 0.002777777777777778;
        double r98007 = 6.0;
        double r98008 = pow(r98005, r98007);
        double r98009 = 0.08333333333333333;
        double r98010 = 4.0;
        double r98011 = pow(r98005, r98010);
        double r98012 = r98009 * r98011;
        double r98013 = fma(r98006, r98008, r98012);
        double r98014 = fma(r98005, r98005, r98013);
        return r98014;
}

Error

Bits error versus x

Target

Original29.3
Target0.1
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.3

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019362 +o rules:numerics
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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