expax (section 3.5)

Average Error: 29.3 → 0.3

Time: 18.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.07174886411196652880040147692852769978344:\\ \;\;\;\;\frac{\log \left(e^{e^{a \cdot \left(3 \cdot x\right)} - 1 \cdot \left(1 \cdot 1\right)}\right)}{\left(1 \cdot e^{a \cdot x} + 1 \cdot 1\right) + e^{a \cdot x} \cdot e^{a \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x + \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)\right) + \frac{1}{6} \cdot \left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right)\\ \end{array}\]

C

double f(double a, double x) {
        double r4743113 = a;
        double r4743114 = x;
        double r4743115 = r4743113 * r4743114;
        double r4743116 = exp(r4743115);
        double r4743117 = 1.0;
        double r4743118 = r4743116 - r4743117;
        return r4743118;
}

↓

double f(double a, double x) {
        double r4743119 = a;
        double r4743120 = x;
        double r4743121 = r4743119 * r4743120;
        double r4743122 = -0.07174886411196653;
        bool r4743123 = r4743121 <= r4743122;
        double r4743124 = 3.0;
        double r4743125 = r4743124 * r4743120;
        double r4743126 = r4743119 * r4743125;
        double r4743127 = exp(r4743126);
        double r4743128 = 1.0;
        double r4743129 = r4743128 * r4743128;
        double r4743130 = r4743128 * r4743129;
        double r4743131 = r4743127 - r4743130;
        double r4743132 = exp(r4743131);
        double r4743133 = log(r4743132);
        double r4743134 = exp(r4743121);
        double r4743135 = r4743128 * r4743134;
        double r4743136 = r4743135 + r4743129;
        double r4743137 = r4743134 * r4743134;
        double r4743138 = r4743136 + r4743137;
        double r4743139 = r4743133 / r4743138;
        double r4743140 = 0.5;
        double r4743141 = r4743140 * r4743121;
        double r4743142 = r4743141 * r4743121;
        double r4743143 = r4743121 + r4743142;
        double r4743144 = 0.16666666666666666;
        double r4743145 = r4743121 * r4743121;
        double r4743146 = r4743121 * r4743145;
        double r4743147 = r4743144 * r4743146;
        double r4743148 = r4743143 + r4743147;
        double r4743149 = r4743123 ? r4743139 : r4743148;
        return r4743149;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.3
Target	0.2
Herbie	0.3

\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (* a x) < -0.07174886411196653

   1. Initial program 0.0

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

   3. Applied flip3-- 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

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

   6. Applied add-log-exp 0.0

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

   7. Applied add-log-exp 0.0

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

   8. Applied diff-log 0.0

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

   9. Simplified 0.0

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

   if -0.07174886411196653 < (* a x)

   1. Initial program 44.2

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

   2. Taylor expanded around 0 13.8

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(a \cdot x + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\right)}\]

   3. Simplified 0.5

      \[\leadsto \color{blue}{\left(\left(x \cdot a\right) \cdot \left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right)\right) \cdot \frac{1}{6} + \left(x \cdot a + \left(x \cdot a\right) \cdot \left(\left(x \cdot a\right) \cdot \frac{1}{2}\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.07174886411196652880040147692852769978344:\\ \;\;\;\;\frac{\log \left(e^{e^{a \cdot \left(3 \cdot x\right)} - 1 \cdot \left(1 \cdot 1\right)}\right)}{\left(1 \cdot e^{a \cdot x} + 1 \cdot 1\right) + e^{a \cdot x} \cdot e^{a \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x + \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)\right) + \frac{1}{6} \cdot \left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (a x)
  :name "expax (section 3.5)"
  :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))