exp2 (problem 3.3.7)

Average Error: 29.4 → 0.6

Time: 7.4s

Precision: binary64

Math

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

↓

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

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (+
  (* 0.08333333333333333 (pow x 4.0))
  (+
   (fma 0.002777777777777778 (pow x 6.0) (* 4.96031746031746e-5 (pow x 8.0)))
   (pow x 2.0))))

C

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

↓

double code(double x) {
	return (0.08333333333333333 * pow(x, 4.0)) + (fma(0.002777777777777778, pow(x, 6.0), (4.96031746031746e-5 * pow(x, 8.0))) + pow(x, 2.0));
}

Error

Bits error versus x

Target

Original	29.4
Target	0.0
Herbie	0.6

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

Derivation

1. Initial program 29.4

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

2. Taylor expanded in x around 0 0.6

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

3. Applied associate-+r+_binary64 0.6

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

4. Simplified 0.6

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

5. Final simplification 0.6

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

Reproduce

herbie shell --seed 2021215 
(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))))