Average Error: 29.8 → 0.7
Time: 28.1s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right), \frac{1}{360}, \mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right), \frac{1}{12}, x \cdot x\right)\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right), \frac{1}{360}, \mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right), \frac{1}{12}, x \cdot x\right)\right)
double f(double x) {
        double r2795143 = x;
        double r2795144 = exp(r2795143);
        double r2795145 = 2.0;
        double r2795146 = r2795144 - r2795145;
        double r2795147 = -r2795143;
        double r2795148 = exp(r2795147);
        double r2795149 = r2795146 + r2795148;
        return r2795149;
}
↓
double f(double x) {
        double r2795150 = x;
        double r2795151 = r2795150 * r2795150;
        double r2795152 = r2795151 * r2795151;
        double r2795153 = r2795152 * r2795151;
        double r2795154 = 0.002777777777777778;
        double r2795155 = 0.08333333333333333;
        double r2795156 = fma(r2795152, r2795155, r2795151);
        double r2795157 = fma(r2795153, r2795154, r2795156);
        return r2795157;
}

Error

Bits error versus x

Target

Original29.8
Target0.0
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.8

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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