expax (section 3.5)

Average Error: 29.6 → 0.7

Time: 2.5s

Precision: binary64

Math

\[e^{a \cdot x} - 1\]

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \leq -6083.193742126296:\\ \;\;\;\;\frac{{\left(e^{\left(a \cdot x\right) \cdot 2}\right)}^{3} + -1}{\left({\left(e^{a \cdot x}\right)}^{4} + \left(e^{\left(a \cdot x\right) \cdot 2} + 1\right)\right) \cdot \left(e^{a \cdot x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(a \cdot x\right)}^{3} \cdot 1.3333333333333333 + 2 \cdot \left(a \cdot x + {\left(a \cdot x\right)}^{2}\right)}{e^{a \cdot x} + 1}\\ \end{array}\]

FPCore

(FPCore (a x) :precision binary64 (- (exp (* a x)) 1.0))

↓

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -6083.193742126296)
   (/
    (+ (pow (exp (* (* a x) 2.0)) 3.0) -1.0)
    (*
     (+ (pow (exp (* a x)) 4.0) (+ (exp (* (* a x) 2.0)) 1.0))
     (+ (exp (* a x)) 1.0)))
   (/
    (+
     (* (pow (* a x) 3.0) 1.3333333333333333)
     (* 2.0 (+ (* a x) (pow (* a x) 2.0))))
    (+ (exp (* a x)) 1.0))))

C

double code(double a, double x) {
	return exp(a * x) - 1.0;
}

↓

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -6083.193742126296) {
		tmp = (pow(exp((a * x) * 2.0), 3.0) + -1.0) / ((pow(exp(a * x), 4.0) + (exp((a * x) * 2.0) + 1.0)) * (exp(a * x) + 1.0));
	} else {
		tmp = ((pow((a * x), 3.0) * 1.3333333333333333) + (2.0 * ((a * x) + pow((a * x), 2.0)))) / (exp(a * x) + 1.0);
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.6
Target	0.2
Herbie	0.7

\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| < 0.1:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -6083.1937421262955

   1. Initial program 0

      \[e^{a \cdot x} - 1\]
   2. Using strategy rm

   3. Applied flip--_binary64 0

      \[\leadsto \color{blue}{\frac{e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1}{e^{a \cdot x} + 1}}\]

   4. Simplified 0

      \[\leadsto \frac{\color{blue}{{\left(e^{a \cdot x}\right)}^{2} + -1}}{e^{a \cdot x} + 1}\]
   5. Using strategy rm

   6. Applied pow-exp_binary64 0

      \[\leadsto \frac{\color{blue}{e^{\left(a \cdot x\right) \cdot 2}} + -1}{e^{a \cdot x} + 1}\]
   7. Using strategy rm

   8. Applied flip3-+_binary64 0

      \[\leadsto \frac{\color{blue}{\frac{{\left(e^{\left(a \cdot x\right) \cdot 2}\right)}^{3} + {-1}^{3}}{e^{\left(a \cdot x\right) \cdot 2} \cdot e^{\left(a \cdot x\right) \cdot 2} + \left(-1 \cdot -1 - e^{\left(a \cdot x\right) \cdot 2} \cdot -1\right)}}}{e^{a \cdot x} + 1}\]

   9. Applied associate-/l/_binary64 0

      \[\leadsto \color{blue}{\frac{{\left(e^{\left(a \cdot x\right) \cdot 2}\right)}^{3} + {-1}^{3}}{\left(e^{a \cdot x} + 1\right) \cdot \left(e^{\left(a \cdot x\right) \cdot 2} \cdot e^{\left(a \cdot x\right) \cdot 2} + \left(-1 \cdot -1 - e^{\left(a \cdot x\right) \cdot 2} \cdot -1\right)\right)}}\]

   10. Simplified 0

       \[\leadsto \frac{{\left(e^{\left(a \cdot x\right) \cdot 2}\right)}^{3} + {-1}^{3}}{\color{blue}{\left({\left(e^{a \cdot x}\right)}^{4} + \left(e^{\left(a \cdot x\right) \cdot 2} + 1\right)\right) \cdot \left(1 + e^{a \cdot x}\right)}}\]

   if -6083.1937421262955 < (*.f64 a x)

   1. Initial program 43.8

      \[e^{a \cdot x} - 1\]
   2. Using strategy rm

   3. Applied flip--_binary64 43.9

      \[\leadsto \color{blue}{\frac{e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1}{e^{a \cdot x} + 1}}\]

   4. Simplified 43.9

      \[\leadsto \frac{\color{blue}{{\left(e^{a \cdot x}\right)}^{2} + -1}}{e^{a \cdot x} + 1}\]
   5. Using strategy rm

   6. Applied pow-exp_binary64 43.8

      \[\leadsto \frac{\color{blue}{e^{\left(a \cdot x\right) \cdot 2}} + -1}{e^{a \cdot x} + 1}\]

   7. Taylor expanded around 0 15.0

      \[\leadsto \frac{\color{blue}{2 \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(1.3333333333333333 \cdot \left({a}^{3} \cdot {x}^{3}\right) + 2 \cdot \left(a \cdot x\right)\right)}}{e^{a \cdot x} + 1}\]

   8. Simplified 1.1

      \[\leadsto \frac{\color{blue}{{\left(a \cdot x\right)}^{3} \cdot 1.3333333333333333 + \left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot 2 + 2\right)}}{e^{a \cdot x} + 1}\]
   9. Using strategy rm

   10. Applied distribute-lft1-in_binary64 1.1

       \[\leadsto \frac{{\left(a \cdot x\right)}^{3} \cdot 1.3333333333333333 + \left(a \cdot x\right) \cdot \color{blue}{\left(\left(a \cdot x + 1\right) \cdot 2\right)}}{e^{a \cdot x} + 1}\]

   11. Applied associate-*r*_binary64 1.1

       \[\leadsto \frac{{\left(a \cdot x\right)}^{3} \cdot 1.3333333333333333 + \color{blue}{\left(\left(a \cdot x\right) \cdot \left(a \cdot x + 1\right)\right) \cdot 2}}{e^{a \cdot x} + 1}\]

   12. Simplified 1.1

       \[\leadsto \frac{{\left(a \cdot x\right)}^{3} \cdot 1.3333333333333333 + \color{blue}{\left(a \cdot x + {\left(a \cdot x\right)}^{2}\right)} \cdot 2}{e^{a \cdot x} + 1}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -6083.193742126296:\\ \;\;\;\;\frac{{\left(e^{\left(a \cdot x\right) \cdot 2}\right)}^{3} + -1}{\left({\left(e^{a \cdot x}\right)}^{4} + \left(e^{\left(a \cdot x\right) \cdot 2} + 1\right)\right) \cdot \left(e^{a \cdot x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(a \cdot x\right)}^{3} \cdot 1.3333333333333333 + 2 \cdot \left(a \cdot x + {\left(a \cdot x\right)}^{2}\right)}{e^{a \cdot x} + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020231 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1.0 (+ (/ (* a x) 2.0) (/ (pow (* a x) 2.0) 6.0)))) (- (exp (* a x)) 1.0))

  (- (exp (* a x)) 1.0))