Average Error: 29.3 → 0.7
Time: 3.7s
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 r79085 = x;
        double r79086 = exp(r79085);
        double r79087 = 2.0;
        double r79088 = r79086 - r79087;
        double r79089 = -r79085;
        double r79090 = exp(r79089);
        double r79091 = r79088 + r79090;
        return r79091;
}
↓
double f(double x) {
        double r79092 = x;
        double r79093 = 0.002777777777777778;
        double r79094 = 6.0;
        double r79095 = pow(r79092, r79094);
        double r79096 = 0.08333333333333333;
        double r79097 = 4.0;
        double r79098 = pow(r79092, r79097);
        double r79099 = r79096 * r79098;
        double r79100 = fma(r79093, r79095, r79099);
        double r79101 = fma(r79092, r79092, r79100);
        return r79101;
}

Error

Bits error versus x

Target

Original29.3
Target0.0
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 2020062 +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))))