exp2 (problem 3.3.7)

Average Error: 29.7 → 0.6

Time: 6.1s

Precision: binary64

Math

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

↓

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

FPCore

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

↓

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

Error

Bits error versus x

Target

Original	29.7
Target	0.0
Herbie	0.6

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

Derivation

1. Initial program 29.7

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

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

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

4. Simplified 0.6

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

5. Final simplification 0.6

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

Reproduce

herbie shell --seed 2022082 
(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))))