expax (section 3.5)

Average Error: 29.8 → 3.4

Time: 5.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \leq -0.012800641208829767:\\ \;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{3} - 1}{1 + e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + x \cdot e^{\log \left(\left(a \cdot a\right) \cdot \left(0.5 + a \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)}\right)\\ \end{array}\]

FPCore

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

↓

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -0.012800641208829767)
   (/
    (- (pow (exp (* a x)) 3.0) 1.0)
    (+ 1.0 (* (exp (* a x)) (+ (exp (* a x)) 1.0))))
   (*
    x
    (+
     a
     (* x (exp (log (* (* a a) (+ 0.5 (* a (* x 0.16666666666666666)))))))))))

C

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

↓

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -0.012800641208829767) {
		tmp = (pow(exp(a * x), 3.0) - 1.0) / (1.0 + (exp(a * x) * (exp(a * x) + 1.0)));
	} else {
		tmp = x * (a + (x * exp(log((a * a) * (0.5 + (a * (x * 0.16666666666666666)))))));
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.8
Target	0.2
Herbie	3.4

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -0.012800641208829767

   1. Initial program 0.0

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

   3. Applied flip3--_binary64_423 0.0

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

   4. Simplified 0.0

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

   5. Simplified 0.0

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

   if -0.012800641208829767 < (*.f64 a x)

   1. Initial program 44.6

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

   2. Taylor expanded around 0 14.8

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left({a}^{3} \cdot {x}^{3}\right) + \left(a \cdot x + 0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right)\right)}\]

   3. Simplified 8.2

      \[\leadsto \color{blue}{x \cdot \left(a + x \cdot \left(0.5 \cdot \left(a \cdot a\right) + x \cdot \left(0.16666666666666666 \cdot {a}^{3}\right)\right)\right)}\]
   4. Using strategy rm

   5. Applied add-exp-log_binary64_457 8.2

      \[\leadsto x \cdot \left(a + x \cdot \color{blue}{e^{\log \left(0.5 \cdot \left(a \cdot a\right) + x \cdot \left(0.16666666666666666 \cdot {a}^{3}\right)\right)}}\right)\]

   6. Simplified 5.0

      \[\leadsto x \cdot \left(a + x \cdot e^{\color{blue}{\log \left(\left(a \cdot a\right) \cdot \left(0.5 + \left(x \cdot 0.16666666666666666\right) \cdot a\right)\right)}}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 3.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -0.012800641208829767:\\ \;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{3} - 1}{1 + e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + x \cdot e^{\log \left(\left(a \cdot a\right) \cdot \left(0.5 + a \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020354 
(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))