Average Error: 29.9 → 0.5
Time: 27.6s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[x \cdot x + \left(x \cdot x\right) \cdot \left(\log \left(e^{\frac{1}{12} \cdot \left(x \cdot x\right)}\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
x \cdot x + \left(x \cdot x\right) \cdot \left(\log \left(e^{\frac{1}{12} \cdot \left(x \cdot x\right)}\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right)
double f(double x) {
        double r2728951 = x;
        double r2728952 = exp(r2728951);
        double r2728953 = 2.0;
        double r2728954 = r2728952 - r2728953;
        double r2728955 = -r2728951;
        double r2728956 = exp(r2728955);
        double r2728957 = r2728954 + r2728956;
        return r2728957;
}
↓
double f(double x) {
        double r2728958 = x;
        double r2728959 = r2728958 * r2728958;
        double r2728960 = 0.08333333333333333;
        double r2728961 = r2728960 * r2728959;
        double r2728962 = exp(r2728961);
        double r2728963 = log(r2728962);
        double r2728964 = r2728959 * r2728959;
        double r2728965 = 0.002777777777777778;
        double r2728966 = r2728964 * r2728965;
        double r2728967 = r2728963 + r2728966;
        double r2728968 = r2728959 * r2728967;
        double r2728969 = r2728959 + r2728968;
        return r2728969;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
Target0.0
Herbie0.5
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.9

    \[\left(e^{x} - 2\right) + e^{-x}\]
  2. Using strategy rm

  3. Applied associate-+l-29.9

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

  4. Taylor expanded around 0 0.5

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

  5. Simplified0.5

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

  7. Applied add-log-exp0.5

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

  8. Final simplification0.5

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

Reproduce

herbie shell --seed 2019146 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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