Average Error: 30.5 → 0.5
Time: 34.8s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[(\left(\log \left(e^{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}}\right)\right) \cdot \left(x \cdot x\right) + \left((\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12} + \left(x \cdot x\right))_*\right))_*\]
\left(e^{x} - 2\right) + e^{-x}
↓
(\left(\log \left(e^{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}}\right)\right) \cdot \left(x \cdot x\right) + \left((\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12} + \left(x \cdot x\right))_*\right))_*
double f(double x) {
        double r6518308 = x;
        double r6518309 = exp(r6518308);
        double r6518310 = 2.0;
        double r6518311 = r6518309 - r6518310;
        double r6518312 = -r6518308;
        double r6518313 = exp(r6518312);
        double r6518314 = r6518311 + r6518313;
        return r6518314;
}
↓
double f(double x) {
        double r6518315 = x;
        double r6518316 = r6518315 * r6518315;
        double r6518317 = r6518316 * r6518316;
        double r6518318 = 0.002777777777777778;
        double r6518319 = r6518317 * r6518318;
        double r6518320 = exp(r6518319);
        double r6518321 = log(r6518320);
        double r6518322 = 0.08333333333333333;
        double r6518323 = fma(r6518317, r6518322, r6518316);
        double r6518324 = fma(r6518321, r6518316, r6518323);
        return r6518324;
}

Error

Bits error versus x

Target

Original30.5
Target0.0
Herbie0.5
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 30.5

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

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

    \[\leadsto \color{blue}{(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right) \cdot \left(x \cdot x\right) + \left((\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12} + \left(x \cdot x\right))_*\right))_*}\]
  4. Using strategy rm

  5. Applied add-log-exp0.5

    \[\leadsto (\color{blue}{\left(\log \left(e^{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}}\right)\right)} \cdot \left(x \cdot x\right) + \left((\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12} + \left(x \cdot x\right))_*\right))_*\]

  6. Final simplification0.5

    \[\leadsto (\left(\log \left(e^{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}}\right)\right) \cdot \left(x \cdot x\right) + \left((\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12} + \left(x \cdot x\right))_*\right))_*\]

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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