exp2 (problem 3.3.7)

Average Error: 29.5 → 0.6

Time: 10.3s

Precision: binary64

Cost: 20352

Math

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

↓

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

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (+
  (* 0.002777777777777778 (pow x 6.0))
  (+
   (+ (* 0.08333333333333333 (pow x 4.0)) (* x x))
   (* 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)) + (((0.08333333333333333 * pow(x, 4.0)) + (x * x)) + (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)) + (((0.08333333333333333d0 * (x ** 4.0d0)) + (x * x)) + (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)) + (((0.08333333333333333 * Math.pow(x, 4.0)) + (x * x)) + (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)) + (((0.08333333333333333 * math.pow(x, 4.0)) + (x * x)) + (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(Float64(Float64(0.08333333333333333 * (x ^ 4.0)) + Float64(x * x)) + 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)) + (((0.08333333333333333 * (x ^ 4.0)) + (x * x)) + (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[(N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(4.96031746031746e-5 * N[Power[x, 8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	29.5
Target	0.0
Herbie	0.6

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

Derivation

1. Initial program 29.5

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

2. Simplified 29.5

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

   [Start] 29.5	\[ \left(e^{x} - 2\right) + e^{-x} \]
   associate-+l- [=>] 29.5	\[ \color{blue}{e^{x} - \left(2 - e^{-x}\right)} \]
   sub-neg [=>] 29.5	\[ \color{blue}{e^{x} + \left(-\left(2 - e^{-x}\right)\right)} \]
   +-commutative [=>] 29.5	\[ \color{blue}{\left(-\left(2 - e^{-x}\right)\right) + e^{x}} \]
   neg-sub0 [=>] 29.5	\[ \color{blue}{\left(0 - \left(2 - e^{-x}\right)\right)} + e^{x} \]
   associate-+l- [=>] 29.5	\[ \color{blue}{0 - \left(\left(2 - e^{-x}\right) - e^{x}\right)} \]
   associate--l- [=>] 29.6	\[ 0 - \color{blue}{\left(2 - \left(e^{-x} + e^{x}\right)\right)} \]
   +-commutative [=>] 29.6	\[ 0 - \left(2 - \color{blue}{\left(e^{x} + e^{-x}\right)}\right) \]
   associate--r+ [=>] 29.5	\[ 0 - \color{blue}{\left(\left(2 - e^{x}\right) - e^{-x}\right)} \]
   associate--r- [=>] 29.5	\[ \color{blue}{\left(0 - \left(2 - e^{x}\right)\right) + e^{-x}} \]
   neg-sub0 [<=] 29.5	\[ \color{blue}{\left(-\left(2 - e^{x}\right)\right)} + e^{-x} \]
   +-commutative [<=] 29.5	\[ \color{blue}{e^{-x} + \left(-\left(2 - e^{x}\right)\right)} \]
   sub-neg [<=] 29.5	\[ \color{blue}{e^{-x} - \left(2 - e^{x}\right)} \]

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. Applied egg-rr 0.6

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

5. Applied egg-rr 0.6

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

6. Final simplification 0.6

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

Alternatives

Alternative 1
Error	0.6
Cost	13632

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

Alternative 2
Error	1.1
Cost	6912

\[0.002777777777777778 \cdot {x}^{6} + x \cdot x \]

Alternative 3
Error	0.8
Cost	6912

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

Alternative 4
Error	1.1
Cost	192

\[x \cdot x \]

Alternative 5
Error	60.2
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2023187 
(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))))