exp2 (problem 3.3.7)

Average Error: 29.6 → 0.5

Time: 7.5s

Precision: binary64

Cost: 19968

Math

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

↓

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

FPCore

(FPCore (x) :precision binary64 (+ (- (exp x) 2.0) (exp (- x))))

↓

(FPCore (x)
 :precision binary64
 (+
  (pow x 2.0)
  (+
   (* 0.08333333333333333 (pow x 4.0))
   (* 0.002777777777777778 (pow x 6.0)))))

C

double code(double x) {
	return (exp(x) - 2.0) + exp(-x);
}

↓

double code(double x) {
	return pow(x, 2.0) + ((0.08333333333333333 * pow(x, 4.0)) + (0.002777777777777778 * pow(x, 6.0)));
}

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 = (x ** 2.0d0) + ((0.08333333333333333d0 * (x ** 4.0d0)) + (0.002777777777777778d0 * (x ** 6.0d0)))
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 Math.pow(x, 2.0) + ((0.08333333333333333 * Math.pow(x, 4.0)) + (0.002777777777777778 * Math.pow(x, 6.0)));
}

Python

def code(x):
	return (math.exp(x) - 2.0) + math.exp(-x)

↓

def code(x):
	return math.pow(x, 2.0) + ((0.08333333333333333 * math.pow(x, 4.0)) + (0.002777777777777778 * math.pow(x, 6.0)))

Julia

function code(x)
	return Float64(Float64(exp(x) - 2.0) + exp(Float64(-x)))
end

↓

function code(x)
	return Float64((x ^ 2.0) + Float64(Float64(0.08333333333333333 * (x ^ 4.0)) + Float64(0.002777777777777778 * (x ^ 6.0))))
end

MATLAB

function tmp = code(x)
	tmp = (exp(x) - 2.0) + exp(-x);
end

↓

function tmp = code(x)
	tmp = (x ^ 2.0) + ((0.08333333333333333 * (x ^ 4.0)) + (0.002777777777777778 * (x ^ 6.0)));
end

Wolfram

code[x_] := N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(N[Power[x, 2.0], $MachinePrecision] + N[(N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.002777777777777778 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	29.6
Target	0.0
Herbie	0.5

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

Derivation

1. Initial program 29.6

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

2. Simplified 29.6

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

   [Start] 29.6	\[ \left(e^{x} - 2\right) + e^{-x} \]
   rational_best-simplify-1 [=>] 29.6	\[ \color{blue}{e^{-x} + \left(e^{x} - 2\right)} \]
   rational_best-simplify-61 [=>] 29.6	\[ e^{-x} + \color{blue}{\left(e^{x} + \left(-2\right)\right)} \]
   rational_best-simplify-115 [=>] 29.6	\[ \color{blue}{e^{x} + \left(\left(-2\right) + e^{-x}\right)} \]
   rational_best-simplify-1 [=>] 29.6	\[ e^{x} + \color{blue}{\left(e^{-x} + \left(-2\right)\right)} \]
   metadata-eval [=>] 29.6	\[ e^{x} + \left(e^{-x} + \color{blue}{-2}\right) \]

3. Taylor expanded in x around 0 0.5

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

4. Simplified 0.5

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

   [Start] 0.5	\[ 0.002777777777777778 \cdot {x}^{6} + \left({x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right) \]
   rational_best-simplify-115 [=>] 0.5	\[ \color{blue}{{x}^{2} + \left(0.08333333333333333 \cdot {x}^{4} + 0.002777777777777778 \cdot {x}^{6}\right)} \]

5. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.7
Cost	13248

\[{x}^{2} + 0.08333333333333333 \cdot {x}^{4} \]

Alternative 2
Error	1.0
Cost	6528

\[{x}^{2} \]

Alternative 3
Error	60.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023104 
(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))))