exp2 (problem 3.3.7)

Average Error: 28.6 → 0.6

Time: 7.9s

Precision: binary64

Cost: 13632

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	28.6
Target	0.0
Herbie	0.6

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

Derivation

1. Initial program 28.6

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

2. Simplified 28.6

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

   [Start] 28.6	\[ \left(e^{x} - 2\right) + e^{-x} \]
   associate-+l- [=>] 28.6	\[ \color{blue}{e^{x} - \left(2 - e^{-x}\right)} \]
   sub-neg [=>] 28.6	\[ \color{blue}{e^{x} + \left(-\left(2 - e^{-x}\right)\right)} \]
   neg-sub0 [=>] 28.6	\[ e^{x} + \color{blue}{\left(0 - \left(2 - e^{-x}\right)\right)} \]
   associate--r- [=>] 28.6	\[ e^{x} + \color{blue}{\left(\left(0 - 2\right) + e^{-x}\right)} \]
   metadata-eval [=>] 28.6	\[ e^{x} + \left(\color{blue}{-2} + e^{-x}\right) \]
   metadata-eval [<=] 28.6	\[ e^{x} + \left(\color{blue}{\left(-2\right)} + e^{-x}\right) \]
   +-commutative [=>] 28.6	\[ e^{x} + \color{blue}{\left(e^{-x} + \left(-2\right)\right)} \]
   metadata-eval [=>] 28.6	\[ e^{x} + \left(e^{-x} + \color{blue}{-2}\right) \]

3. Taylor expanded in x around 0 0.6

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

4. Simplified 0.6

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

   [Start] 0.6	\[ 0.002777777777777778 \cdot {x}^{6} + \left({x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right) \]
   fma-def [=>] 0.6	\[ \color{blue}{\mathsf{fma}\left(0.002777777777777778, {x}^{6}, {x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right)} \]
   unpow2 [=>] 0.6	\[ \mathsf{fma}\left(0.002777777777777778, {x}^{6}, \color{blue}{x \cdot x} + 0.08333333333333333 \cdot {x}^{4}\right) \]

5. Applied egg-rr 0.6

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

6. Final simplification 0.6

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

Alternatives

Alternative 1
Error	0.7
Cost	6912

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

Alternative 2
Error	1.1
Cost	192

\[x \cdot x \]

Alternative 3
Error	60.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023016 
(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))))