Average Error: 29.3 → 0.6
Time: 1.1m
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\mathsf{fma}\left(\left(\log \left(e^{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}}\right)\right), \left(x \cdot x\right), \left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \frac{1}{12}, \left(x \cdot x\right)\right)\right)\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
\mathsf{fma}\left(\left(\log \left(e^{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}}\right)\right), \left(x \cdot x\right), \left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \frac{1}{12}, \left(x \cdot x\right)\right)\right)\right)
double f(double x) {
        double r6813345 = x;
        double r6813346 = exp(r6813345);
        double r6813347 = 2.0;
        double r6813348 = r6813346 - r6813347;
        double r6813349 = -r6813345;
        double r6813350 = exp(r6813349);
        double r6813351 = r6813348 + r6813350;
        return r6813351;
}
↓
double f(double x) {
        double r6813352 = x;
        double r6813353 = r6813352 * r6813352;
        double r6813354 = r6813353 * r6813353;
        double r6813355 = 0.002777777777777778;
        double r6813356 = r6813354 * r6813355;
        double r6813357 = exp(r6813356);
        double r6813358 = log(r6813357);
        double r6813359 = 0.08333333333333333;
        double r6813360 = fma(r6813354, r6813359, r6813353);
        double r6813361 = fma(r6813358, r6813353, r6813360);
        return r6813361;
}

Error

Bits error versus x

Target

Original29.3
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.3

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

  2. Simplified29.3

    \[\leadsto \color{blue}{\left(e^{x} - 2\right) - \frac{-1}{e^{x}}}\]

  3. Taylor expanded around 0 0.6

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

  4. Simplified0.6

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

  6. Applied add-log-exp0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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