Average Error: 29.6 → 0.6
Time: 8.2s
Precision: binary64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\left(x \cdot x + \left(0.08333333333333333 \cdot {x}^{4} + 0.002777777777777778 \cdot {x}^{6}\right)\right) + 4.96031746031746 \cdot 10^{-05} \cdot {x}^{8}\]
\left(e^{x} - 2\right) + e^{-x}
↓
\left(x \cdot x + \left(0.08333333333333333 \cdot {x}^{4} + 0.002777777777777778 \cdot {x}^{6}\right)\right) + 4.96031746031746 \cdot 10^{-05} \cdot {x}^{8}
(FPCore (x) :precision binary64 (+ (- (exp x) 2.0) (exp (- x))))
↓
(FPCore (x)
 :precision binary64
 (+
  (+
   (* x x)
   (+
    (* 0.08333333333333333 (pow x 4.0))
    (* 0.002777777777777778 (pow x 6.0))))
  (* 4.96031746031746e-05 (pow x 8.0))))
double code(double x) {
	return (exp(x) - 2.0) + exp(-x);
}
↓
double code(double x) {
	return ((x * x) + ((0.08333333333333333 * pow(x, 4.0)) + (0.002777777777777778 * pow(x, 6.0)))) + (4.96031746031746e-05 * pow(x, 8.0));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 29.6

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

  2. Taylor expanded around 0 0.6

    \[\leadsto \color{blue}{4.96031746031746 \cdot 10^{-05} \cdot {x}^{8} + \left({x}^{2} + \left(0.002777777777777778 \cdot {x}^{6} + 0.08333333333333333 \cdot {x}^{4}\right)\right)}\]

  3. Simplified0.6

    \[\leadsto \color{blue}{\left(x \cdot x + \left(0.08333333333333333 \cdot {x}^{4} + 0.002777777777777778 \cdot {x}^{6}\right)\right) + 4.96031746031746 \cdot 10^{-05} \cdot {x}^{8}}\]

  4. Final simplification0.6

    \[\leadsto \left(x \cdot x + \left(0.08333333333333333 \cdot {x}^{4} + 0.002777777777777778 \cdot {x}^{6}\right)\right) + 4.96031746031746 \cdot 10^{-05} \cdot {x}^{8}\]

Reproduce

herbie shell --seed 2021050 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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