Average Error: 29.6 → 0.6
Time: 9.8s
Precision: binary64
Cost: 26688
\[\left(e^{x} - 2\right) + e^{-x} \]
↓
\[0.002777777777777778 \cdot {x}^{6} + \left({x}^{2} + \left(0.08333333333333333 \cdot {x}^{4} + 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8}\right)\right) \]
(FPCore (x) :precision binary64 (+ (- (exp x) 2.0) (exp (- x))))
↓
(FPCore (x)
 :precision binary64
 (+
  (* 0.002777777777777778 (pow x 6.0))
  (+
   (pow x 2.0)
   (+
    (* 0.08333333333333333 (pow x 4.0))
    (* 4.96031746031746e-5 (pow x 8.0))))))
double code(double x) {
	return (exp(x) - 2.0) + exp(-x);
}
↓
double code(double x) {
	return (0.002777777777777778 * pow(x, 6.0)) + (pow(x, 2.0) + ((0.08333333333333333 * pow(x, 4.0)) + (4.96031746031746e-5 * pow(x, 8.0))));
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = (exp(x) - 2.0d0) + exp(-x)
end function
↓
real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.002777777777777778d0 * (x ** 6.0d0)) + ((x ** 2.0d0) + ((0.08333333333333333d0 * (x ** 4.0d0)) + (4.96031746031746d-5 * (x ** 8.0d0))))
end function

public static double code(double x) {
	return (Math.exp(x) - 2.0) + Math.exp(-x);
}
↓
public static double code(double x) {
	return (0.002777777777777778 * Math.pow(x, 6.0)) + (Math.pow(x, 2.0) + ((0.08333333333333333 * Math.pow(x, 4.0)) + (4.96031746031746e-5 * Math.pow(x, 8.0))));
}

def code(x):
	return (math.exp(x) - 2.0) + math.exp(-x)
↓
def code(x):
	return (0.002777777777777778 * math.pow(x, 6.0)) + (math.pow(x, 2.0) + ((0.08333333333333333 * math.pow(x, 4.0)) + (4.96031746031746e-5 * math.pow(x, 8.0))))

function code(x)
	return Float64(Float64(exp(x) - 2.0) + exp(Float64(-x)))
end
↓
function code(x)
	return Float64(Float64(0.002777777777777778 * (x ^ 6.0)) + Float64((x ^ 2.0) + Float64(Float64(0.08333333333333333 * (x ^ 4.0)) + Float64(4.96031746031746e-5 * (x ^ 8.0)))))
end

function tmp = code(x)
	tmp = (exp(x) - 2.0) + exp(-x);
end
↓
function tmp = code(x)
	tmp = (0.002777777777777778 * (x ^ 6.0)) + ((x ^ 2.0) + ((0.08333333333333333 * (x ^ 4.0)) + (4.96031746031746e-5 * (x ^ 8.0))));
end

code[x_] := N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]
↓
code[x_] := N[(N[(0.002777777777777778 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, 2.0], $MachinePrecision] + N[(N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(4.96031746031746e-5 * N[Power[x, 8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

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. Simplified29.6

    \[\leadsto \color{blue}{-2 + \left(e^{x} + e^{-x}\right)} \]
    Proof
    (+.f64 -2 (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite<= metadata-eval (neg.f64 2)) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (neg.f64 2) (exp.f64 x)) (exp.f64 (neg.f64 x)))): 2 points increase in error, 2 points decrease in error
    (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (exp.f64 x) (neg.f64 2))) (exp.f64 (neg.f64 x))): 0 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite<= sub-neg_binary64 (-.f64 (exp.f64 x) 2)) (exp.f64 (neg.f64 x))): 0 points increase in error, 0 points decrease in error

  3. Taylor expanded in x around 0 0.6

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

  4. Final simplification0.6

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

Alternatives

Alternative 1
Error0.6
Cost19968
\[0.002777777777777778 \cdot {x}^{6} + \left({x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right) \]
Alternative 2
Error0.7
Cost13184
\[\mathsf{fma}\left(0.08333333333333333, {x}^{4}, x \cdot x\right) \]
Alternative 3
Error1.0
Cost192
\[x \cdot x \]

Error

Reproduce

herbie shell --seed 2022316 
(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))))