Average Error: 29.4 → 0.6
Time: 17.5s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[x \cdot x + \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \frac{1}{12}, \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
x \cdot x + \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \frac{1}{12}, \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)
double f(double x) {
        double r1876183 = x;
        double r1876184 = exp(r1876183);
        double r1876185 = 2.0;
        double r1876186 = r1876184 - r1876185;
        double r1876187 = -r1876183;
        double r1876188 = exp(r1876187);
        double r1876189 = r1876186 + r1876188;
        return r1876189;
}
↓
double f(double x) {
        double r1876190 = x;
        double r1876191 = r1876190 * r1876190;
        double r1876192 = r1876191 * r1876191;
        double r1876193 = 0.08333333333333333;
        double r1876194 = r1876190 * r1876191;
        double r1876195 = 0.002777777777777778;
        double r1876196 = r1876194 * r1876195;
        double r1876197 = r1876196 * r1876194;
        double r1876198 = fma(r1876192, r1876193, r1876197);
        double r1876199 = r1876191 + r1876198;
        return r1876199;
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 29.4

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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