Average Error: 29.9 → 0.6
Time: 5.5s
Precision: binary64
Cost: 13632
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[x \cdot x + \left(0.08333333333333333 \cdot {x}^{4} + 0.002777777777777778 \cdot {x}^{6}\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
x \cdot x + \left(0.08333333333333333 \cdot {x}^{4} + 0.002777777777777778 \cdot {x}^{6}\right)
(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)))))
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)));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
Target0.0
Herbie0.6
\[4 \cdot {\sinh \left(\frac{x}{2}\right)}^{2}\]

Alternatives

Alternative 1
Error0.8
Cost6912
\[x \cdot x + 0.08333333333333333 \cdot {x}^{4}\]
Alternative 2
Error1.1
Cost192
\[x \cdot x\]
Alternative 3
Error31.5
Cost64
\[0\]
Alternative 4
Error61.2
Cost64
\[1\]

Error

Derivation

  1. Initial program 29.9

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Simplified0.6

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

  5. Final simplification0.6

    \[\leadsto x \cdot x + \left(0.08333333333333333 \cdot {x}^{4} + 0.002777777777777778 \cdot {x}^{6}\right)\]

Reproduce

herbie shell --seed 2021044 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64
  :herbie-expected 1.5

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

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