Average Error: 29.1 → 0.5
Time: 6.5s
Precision: binary64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[4.96031746031746 \cdot 10^{-05} \cdot {x}^{8} + \left({x}^{2} + \left(0.002777777777777778 \cdot {x}^{6} + 0.08333333333333333 \cdot {x}^{4}\right)\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
4.96031746031746 \cdot 10^{-05} \cdot {x}^{8} + \left({x}^{2} + \left(0.002777777777777778 \cdot {x}^{6} + 0.08333333333333333 \cdot {x}^{4}\right)\right)
(FPCore (x) :precision binary64 (+ (- (exp x) 2.0) (exp (- x))))
↓
(FPCore (x)
 :precision binary64
 (+
  (* 4.96031746031746e-05 (pow x 8.0))
  (+
   (pow x 2.0)
   (+
    (* 0.002777777777777778 (pow x 6.0))
    (* 0.08333333333333333 (pow x 4.0))))))
double code(double x) {
	return (exp(x) - 2.0) + exp(-x);
}
↓
double code(double x) {
	return (4.96031746031746e-05 * pow(x, 8.0)) + (pow(x, 2.0) + ((0.002777777777777778 * pow(x, 6.0)) + (0.08333333333333333 * pow(x, 4.0))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.1
Target0.0
Herbie0.5
\[4 \cdot {\sinh \left(\frac{x}{2}\right)}^{2}\]

Derivation

  1. Initial program 29.1

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

  2. Taylor expanded around 0 0.5

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

  3. Final simplification0.5

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

Reproduce

herbie shell --seed 2021174 
(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))))