expax (section 3.5)

Average Error: 29.1 → 13.7

Time: 18.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -5.0289586708105234 \cdot 10^{+104}:\\ \;\;\;\;\frac{\frac{-1 + e^{\left(x + \left(x + x\right)\right) \cdot \left(3 \cdot a\right)}}{e^{\left(x \cdot a\right) \cdot 3} \cdot \left(e^{\left(x \cdot a\right) \cdot 3} + 1\right) + 1}}{e^{x \cdot a} \cdot \left(e^{x \cdot a} + 1\right) + 1}\\ \mathbf{elif}\;a \le 4.1836874807488166 \cdot 10^{+54}:\\ \;\;\;\;\frac{\left(x \cdot a\right) \cdot 3 + \frac{9}{2} \cdot \left(x \cdot \left(\left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right) \cdot a\right) + \left(x \cdot a\right) \cdot \left(x \cdot a\right)\right)}{e^{x \cdot a} \cdot \left(e^{x \cdot a} + 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 + e^{\left(x + \left(x + x\right)\right) \cdot \left(3 \cdot a\right)}}{e^{\left(x \cdot a\right) \cdot 3} \cdot \left(e^{\left(x \cdot a\right) \cdot 3} + 1\right) + 1}}{e^{x \cdot a} \cdot \left(e^{x \cdot a} + 1\right) + 1}\\ \end{array}\]

C

double f(double a, double x) {
        double r4728246 = a;
        double r4728247 = x;
        double r4728248 = r4728246 * r4728247;
        double r4728249 = exp(r4728248);
        double r4728250 = 1.0;
        double r4728251 = r4728249 - r4728250;
        return r4728251;
}

↓

double f(double a, double x) {
        double r4728252 = a;
        double r4728253 = -5.0289586708105234e+104;
        bool r4728254 = r4728252 <= r4728253;
        double r4728255 = -1.0;
        double r4728256 = x;
        double r4728257 = r4728256 + r4728256;
        double r4728258 = r4728256 + r4728257;
        double r4728259 = 3.0;
        double r4728260 = r4728259 * r4728252;
        double r4728261 = r4728258 * r4728260;
        double r4728262 = exp(r4728261);
        double r4728263 = r4728255 + r4728262;
        double r4728264 = r4728256 * r4728252;
        double r4728265 = r4728264 * r4728259;
        double r4728266 = exp(r4728265);
        double r4728267 = 1.0;
        double r4728268 = r4728266 + r4728267;
        double r4728269 = r4728266 * r4728268;
        double r4728270 = r4728269 + r4728267;
        double r4728271 = r4728263 / r4728270;
        double r4728272 = exp(r4728264);
        double r4728273 = r4728272 + r4728267;
        double r4728274 = r4728272 * r4728273;
        double r4728275 = r4728274 + r4728267;
        double r4728276 = r4728271 / r4728275;
        double r4728277 = 4.1836874807488166e+54;
        bool r4728278 = r4728252 <= r4728277;
        double r4728279 = 4.5;
        double r4728280 = r4728264 * r4728264;
        double r4728281 = r4728280 * r4728252;
        double r4728282 = r4728256 * r4728281;
        double r4728283 = r4728282 + r4728280;
        double r4728284 = r4728279 * r4728283;
        double r4728285 = r4728265 + r4728284;
        double r4728286 = r4728285 / r4728275;
        double r4728287 = r4728278 ? r4728286 : r4728276;
        double r4728288 = r4728254 ? r4728276 : r4728287;
        return r4728288;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.1
Target	0.2
Herbie	13.7

\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if a < -5.0289586708105234e+104 or 4.1836874807488166e+54 < a

   1. Initial program 16.7

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

   3. Applied flip3-- 16.8

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

   4. Simplified 16.7

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

   5. Simplified 16.7

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

   7. Applied flip3-- 16.7

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

   8. Simplified 16.6

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

   9. Simplified 16.6

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

   if -5.0289586708105234e+104 < a < 4.1836874807488166e+54

   1. Initial program 34.1

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

   3. Applied flip3-- 34.1

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

   4. Simplified 34.0

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

   5. Simplified 34.0

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

   6. Taylor expanded around 0 19.5

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

   7. Simplified 12.5

      \[\leadsto \frac{\color{blue}{\frac{9}{2} \cdot \left(x \cdot \left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot a\right) + \left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) + \left(a \cdot x\right) \cdot 3}}{1 + e^{x \cdot a} \cdot \left(e^{x \cdot a} + 1\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 13.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -5.0289586708105234 \cdot 10^{+104}:\\ \;\;\;\;\frac{\frac{-1 + e^{\left(x + \left(x + x\right)\right) \cdot \left(3 \cdot a\right)}}{e^{\left(x \cdot a\right) \cdot 3} \cdot \left(e^{\left(x \cdot a\right) \cdot 3} + 1\right) + 1}}{e^{x \cdot a} \cdot \left(e^{x \cdot a} + 1\right) + 1}\\ \mathbf{elif}\;a \le 4.1836874807488166 \cdot 10^{+54}:\\ \;\;\;\;\frac{\left(x \cdot a\right) \cdot 3 + \frac{9}{2} \cdot \left(x \cdot \left(\left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right) \cdot a\right) + \left(x \cdot a\right) \cdot \left(x \cdot a\right)\right)}{e^{x \cdot a} \cdot \left(e^{x \cdot a} + 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 + e^{\left(x + \left(x + x\right)\right) \cdot \left(3 \cdot a\right)}}{e^{\left(x \cdot a\right) \cdot 3} \cdot \left(e^{\left(x \cdot a\right) \cdot 3} + 1\right) + 1}}{e^{x \cdot a} \cdot \left(e^{x \cdot a} + 1\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (a x)
  :name "expax (section 3.5)"
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 1/10) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

  (- (exp (* a x)) 1))