Average Error: 29.5 → 0.6
Time: 16.3s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\sqrt{\frac{1}{360} \cdot {x}^{6} + \left(x \cdot x + {x}^{4} \cdot \frac{1}{12}\right)} \cdot \sqrt{\frac{1}{360} \cdot {x}^{6} + \left(x \cdot x + {x}^{4} \cdot \frac{1}{12}\right)}\]
\left(e^{x} - 2\right) + e^{-x}
↓
\sqrt{\frac{1}{360} \cdot {x}^{6} + \left(x \cdot x + {x}^{4} \cdot \frac{1}{12}\right)} \cdot \sqrt{\frac{1}{360} \cdot {x}^{6} + \left(x \cdot x + {x}^{4} \cdot \frac{1}{12}\right)}
double f(double x) {
        double r84056 = x;
        double r84057 = exp(r84056);
        double r84058 = 2.0;
        double r84059 = r84057 - r84058;
        double r84060 = -r84056;
        double r84061 = exp(r84060);
        double r84062 = r84059 + r84061;
        return r84062;
}
↓
double f(double x) {
        double r84063 = 0.002777777777777778;
        double r84064 = x;
        double r84065 = 6.0;
        double r84066 = pow(r84064, r84065);
        double r84067 = r84063 * r84066;
        double r84068 = r84064 * r84064;
        double r84069 = 4.0;
        double r84070 = pow(r84064, r84069);
        double r84071 = 0.08333333333333333;
        double r84072 = r84070 * r84071;
        double r84073 = r84068 + r84072;
        double r84074 = r84067 + r84073;
        double r84075 = sqrt(r84074);
        double r84076 = r84075 * r84075;
        return r84076;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.5
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.5

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

  2. Simplified29.5

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

  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

    \[\leadsto \color{blue}{x \cdot x + \left({x}^{6} \cdot \frac{1}{360} + \frac{1}{12} \cdot {x}^{4}\right)}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt0.6

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

  7. Simplified0.6

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

  8. Simplified0.6

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

  9. Final simplification0.6

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

Reproduce

herbie shell --seed 2019174 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

  :herbie-target
  (* 4.0 (pow (sinh (/ x 2.0)) 2.0))

  (+ (- (exp x) 2.0) (exp (- x))))