exp2 (problem 3.3.7)

Average Error: 29.3 → 0.6

Time: 5.3s

Precision: binary64

Math

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

↓

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

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (fma
  0.08333333333333333
  (pow x 4.0)
  (fma
   0.002777777777777778
   (pow x 6.0)
   (fma 4.96031746031746e-5 (pow x 8.0) (* x x)))))

C

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

↓

double code(double x) {
	return fma(0.08333333333333333, pow(x, 4.0), fma(0.002777777777777778, pow(x, 6.0), fma(4.96031746031746e-5, pow(x, 8.0), (x * x))));
}

Julia

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

↓

function code(x)
	return fma(0.08333333333333333, (x ^ 4.0), fma(0.002777777777777778, (x ^ 6.0), fma(4.96031746031746e-5, (x ^ 8.0), Float64(x * x))))
end

Wolfram

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

↓

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

Error

Bits error versus x

Target

Original	29.3
Target	0.0
Herbie	0.6

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

Derivation

1. Initial program 29.3

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

2. Simplified 29.3

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

3. Taylor expanded in x around 0 0.6

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

4. Simplified 0.6

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

5. Final simplification 0.6

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

Reproduce

herbie shell --seed 2022155 
(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))))