Average Error: 29.4 → 0.5
Time: 4.6s
Precision: binary64
\[\left(e^{x} - 2\right) + e^{-x} \]
↓
\[0.08333333333333333 \cdot {x}^{4} + \left(0.002777777777777778 \cdot {x}^{6} + \left(4.96031746031746 \cdot 10^{-5} \cdot {x}^{8} + {x}^{2}\right)\right) \]
\left(e^{x} - 2\right) + e^{-x}
↓
0.08333333333333333 \cdot {x}^{4} + \left(0.002777777777777778 \cdot {x}^{6} + \left(4.96031746031746 \cdot 10^{-5} \cdot {x}^{8} + {x}^{2}\right)\right)

(FPCore (x) :precision binary64 (+ (- (exp x) 2.0) (exp (- x))))
↓
(FPCore (x)
 :precision binary64
 (+
  (* 0.08333333333333333 (pow x 4.0))
  (+
   (* 0.002777777777777778 (pow x 6.0))
   (+ (* 4.96031746031746e-5 (pow x 8.0)) (pow x 2.0)))))
double code(double x) {
	return (exp(x) - 2.0) + exp(-x);
}
↓
double code(double x) {
	return (0.08333333333333333 * pow(x, 4.0)) + ((0.002777777777777778 * pow(x, 6.0)) + ((4.96031746031746e-5 * pow(x, 8.0)) + pow(x, 2.0)));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 29.4

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

  2. Taylor expanded in x around 0 0.5

    \[\leadsto \color{blue}{0.08333333333333333 \cdot {x}^{4} + \left(0.002777777777777778 \cdot {x}^{6} + \left(4.96031746031746 \cdot 10^{-5} \cdot {x}^{8} + {x}^{2}\right)\right)} \]

  3. Final simplification0.5

    \[\leadsto 0.08333333333333333 \cdot {x}^{4} + \left(0.002777777777777778 \cdot {x}^{6} + \left(4.96031746031746 \cdot 10^{-5} \cdot {x}^{8} + {x}^{2}\right)\right) \]

Reproduce

herbie shell --seed 2022019 
(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))))