Average Error: 29.5 → 0.7
Time: 9.8s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[{x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
{x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)
double f(double x) {
        double r72091 = x;
        double r72092 = exp(r72091);
        double r72093 = 2.0;
        double r72094 = r72092 - r72093;
        double r72095 = -r72091;
        double r72096 = exp(r72095);
        double r72097 = r72094 + r72096;
        return r72097;
}
↓
double f(double x) {
        double r72098 = x;
        double r72099 = 2.0;
        double r72100 = pow(r72098, r72099);
        double r72101 = 0.002777777777777778;
        double r72102 = 6.0;
        double r72103 = pow(r72098, r72102);
        double r72104 = r72101 * r72103;
        double r72105 = 0.08333333333333333;
        double r72106 = 4.0;
        double r72107 = pow(r72098, r72106);
        double r72108 = r72105 * r72107;
        double r72109 = r72104 + r72108;
        double r72110 = r72100 + r72109;
        return r72110;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.5
Target0.0
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.5

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

  2. Taylor expanded around 0 0.7

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

  3. Final simplification0.7

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

Reproduce

herbie shell --seed 2020047 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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