expax (section 3.5)

Average Error: 29.8 → 0.4

Time: 19.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.01289790256637027461572575504078486119397:\\ \;\;\;\;\frac{e^{\left(a \cdot x\right) \cdot 3} - \left(1 \cdot 1\right) \cdot 1}{\mathsf{fma}\left(e^{a \cdot x}, \left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}\right) \cdot \sqrt[3]{e^{a \cdot x}} + 1, 1 \cdot 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \left(a \cdot x\right) \cdot \left(a \cdot x\right), a \cdot x + \left(\frac{1}{6} \cdot a\right) \cdot \left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot x\right)\right)\\ \end{array}\]

C

double f(double a, double x) {
        double r4864380 = a;
        double r4864381 = x;
        double r4864382 = r4864380 * r4864381;
        double r4864383 = exp(r4864382);
        double r4864384 = 1.0;
        double r4864385 = r4864383 - r4864384;
        return r4864385;
}

↓

double f(double a, double x) {
        double r4864386 = a;
        double r4864387 = x;
        double r4864388 = r4864386 * r4864387;
        double r4864389 = -0.012897902566370275;
        bool r4864390 = r4864388 <= r4864389;
        double r4864391 = 3.0;
        double r4864392 = r4864388 * r4864391;
        double r4864393 = exp(r4864392);
        double r4864394 = 1.0;
        double r4864395 = r4864394 * r4864394;
        double r4864396 = r4864395 * r4864394;
        double r4864397 = r4864393 - r4864396;
        double r4864398 = exp(r4864388);
        double r4864399 = cbrt(r4864398);
        double r4864400 = r4864399 * r4864399;
        double r4864401 = r4864400 * r4864399;
        double r4864402 = r4864401 + r4864394;
        double r4864403 = fma(r4864398, r4864402, r4864395);
        double r4864404 = r4864397 / r4864403;
        double r4864405 = 0.5;
        double r4864406 = r4864388 * r4864388;
        double r4864407 = 0.16666666666666666;
        double r4864408 = r4864407 * r4864386;
        double r4864409 = r4864406 * r4864387;
        double r4864410 = r4864408 * r4864409;
        double r4864411 = r4864388 + r4864410;
        double r4864412 = fma(r4864405, r4864406, r4864411);
        double r4864413 = r4864390 ? r4864404 : r4864412;
        return r4864413;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.8
Target	0.1
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if (* a x) < -0.012897902566370275

   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

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

   5. Simplified 0.0

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

   7. Applied add-cube-cbrt 0.0

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

   if -0.012897902566370275 < (* a x)

   1. Initial program 44.7

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

   2. Taylor expanded around 0 15.5

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

   3. Simplified 0.5

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

   4. Taylor expanded around inf 15.5

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

   5. Simplified 0.5

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.01289790256637027461572575504078486119397:\\ \;\;\;\;\frac{e^{\left(a \cdot x\right) \cdot 3} - \left(1 \cdot 1\right) \cdot 1}{\mathsf{fma}\left(e^{a \cdot x}, \left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}\right) \cdot \sqrt[3]{e^{a \cdot x}} + 1, 1 \cdot 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \left(a \cdot x\right) \cdot \left(a \cdot x\right), a \cdot x + \left(\frac{1}{6} \cdot a\right) \cdot \left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(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))