Average Error: 30.0 → 0.6
Time: 1.2m
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[(\left(\log \left(e^{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}}\right)\right) \cdot \left(x \cdot x\right) + \left((\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12} + \left(x \cdot x\right))_*\right))_*\]
double f(double x) {
        double r8855138 = x;
        double r8855139 = exp(r8855138);
        double r8855140 = 2.0;
        double r8855141 = r8855139 - r8855140;
        double r8855142 = -r8855138;
        double r8855143 = exp(r8855142);
        double r8855144 = r8855141 + r8855143;
        return r8855144;
}
↓
double f(double x) {
        double r8855145 = x;
        double r8855146 = r8855145 * r8855145;
        double r8855147 = r8855146 * r8855146;
        double r8855148 = 0.002777777777777778;
        double r8855149 = r8855147 * r8855148;
        double r8855150 = exp(r8855149);
        double r8855151 = log(r8855150);
        double r8855152 = 0.08333333333333333;
        double r8855153 = fma(r8855147, r8855152, r8855146);
        double r8855154 = fma(r8855151, r8855146, r8855153);
        return r8855154;
}


\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))_*

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 30.0

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

  2. Simplified30.0

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

  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

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

  6. Applied add-log-exp0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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