Average Error: 30.1 → 0.6
Time: 11.6s
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 r66937 = x;
        double r66938 = exp(r66937);
        double r66939 = 2.0;
        double r66940 = r66938 - r66939;
        double r66941 = -r66937;
        double r66942 = exp(r66941);
        double r66943 = r66940 + r66942;
        return r66943;
}
↓
double f(double x) {
        double r66944 = x;
        double r66945 = 0.002777777777777778;
        double r66946 = 6.0;
        double r66947 = pow(r66944, r66946);
        double r66948 = 0.08333333333333333;
        double r66949 = 4.0;
        double r66950 = pow(r66944, r66949);
        double r66951 = r66948 * r66950;
        double r66952 = fma(r66945, r66947, r66951);
        double r66953 = fma(r66944, r66944, r66952);
        return r66953;
}

Error

Bits error versus x

Target

Original30.1
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 30.1

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2020042 +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))))