exp2 (problem 3.3.7)

Average Error: 46.58% → 0.71%

Time: 12.2s

Precision: binary64

Cost: 26688

Math

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

↓

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

FPCore

(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))))))

C

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))));
}

Fortran

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

Java

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))));
}

Python

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))))

Julia

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

MATLAB

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

Wolfram

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]

Target

Original	46.58%
Target	0.07%
Herbie	0.71%

\[4 \cdot {\sinh \left(\frac{x}{2}\right)}^{2} \]

Derivation

1. Initial program 46.58

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

2. Simplified 46.59

   \[\leadsto \color{blue}{e^{x} + \left(e^{-x} + -2\right)} \]
   Proof

   [Start] 46.58	\[ \left(e^{x} - 2\right) + e^{-x} \]
   associate-+l- [=>] 46.59	\[ \color{blue}{e^{x} - \left(2 - e^{-x}\right)} \]
   sub-neg [=>] 46.59	\[ \color{blue}{e^{x} + \left(-\left(2 - e^{-x}\right)\right)} \]
   neg-sub0 [=>] 46.59	\[ e^{x} + \color{blue}{\left(0 - \left(2 - e^{-x}\right)\right)} \]
   associate--r- [=>] 46.59	\[ e^{x} + \color{blue}{\left(\left(0 - 2\right) + e^{-x}\right)} \]
   metadata-eval [=>] 46.59	\[ e^{x} + \left(\color{blue}{-2} + e^{-x}\right) \]
   metadata-eval [<=] 46.59	\[ e^{x} + \left(\color{blue}{\left(-2\right)} + e^{-x}\right) \]
   +-commutative [=>] 46.59	\[ e^{x} + \color{blue}{\left(e^{-x} + \left(-2\right)\right)} \]
   metadata-eval [=>] 46.59	\[ e^{x} + \left(e^{-x} + \color{blue}{-2}\right) \]

3. Taylor expanded in x around 0 0.71

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

4. Final simplification 0.71

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

Alternatives

Alternative 1
Error	1.01%
Cost	6912

\[0.08333333333333333 \cdot {x}^{4} + x \cdot x \]

Alternative 2
Error	1.58%
Cost	192

\[x \cdot x \]

Alternative 3
Error	94.1%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023121 
(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))))