expq2 (section 3.11)

Average Error: 40.2 → 0.6

Time: 13.1s

Precision: 64

• could not determine a ground truth for program body

  1. x = 8.049456502996968e+74

Math

\[\frac{e^{x}}{e^{x} - 1}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.0005386962406274384:\\ \;\;\;\;\sqrt[3]{\frac{e^{x}}{\mathsf{expm1}\left(x\right)} \cdot \left(\frac{e^{x}}{\mathsf{expm1}\left(x\right)} \cdot \frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{12}, \left(\frac{1}{2} + \frac{1}{x}\right)\right)\\ \end{array}\]

C

double f(double x) {
        double r5075202 = x;
        double r5075203 = exp(r5075202);
        double r5075204 = 1.0;
        double r5075205 = r5075203 - r5075204;
        double r5075206 = r5075203 / r5075205;
        return r5075206;
}

↓

double f(double x) {
        double r5075207 = x;
        double r5075208 = -0.0005386962406274384;
        bool r5075209 = r5075207 <= r5075208;
        double r5075210 = exp(r5075207);
        double r5075211 = expm1(r5075207);
        double r5075212 = r5075210 / r5075211;
        double r5075213 = r5075212 * r5075212;
        double r5075214 = r5075212 * r5075213;
        double r5075215 = cbrt(r5075214);
        double r5075216 = 0.08333333333333333;
        double r5075217 = 0.5;
        double r5075218 = 1.0;
        double r5075219 = r5075218 / r5075207;
        double r5075220 = r5075217 + r5075219;
        double r5075221 = fma(r5075207, r5075216, r5075220);
        double r5075222 = r5075209 ? r5075215 : r5075221;
        return r5075222;
}

Error

Bits error versus x

Target

Original	40.2
Target	39.8
Herbie	0.6

\[\frac{1}{1 - e^{-x}}\]

Derivation

1. Split input into 2 regimes

2. if x < -0.0005386962406274384

   1. Initial program 0.0

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

   3. Applied add-cbrt-cube 0.0

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

   4. Applied add-cbrt-cube 0.1

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

   5. Applied cbrt-undiv 0.1

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

   6. Simplified 0.1

      \[\leadsto \sqrt[3]{\color{blue}{\frac{e^{x}}{\mathsf{expm1}\left(x\right)} \cdot \left(\frac{e^{x}}{\mathsf{expm1}\left(x\right)} \cdot \frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right)}}\]

   if -0.0005386962406274384 < x

   1. Initial program 60.2

      \[\frac{e^{x}}{e^{x} - 1}\]

   2. Taylor expanded around 0 0.8

      \[\leadsto \color{blue}{\frac{1}{12} \cdot x + \left(\frac{1}{x} + \frac{1}{2}\right)}\]

   3. Simplified 0.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{12}, \left(\frac{1}{2} + \frac{1}{x}\right)\right)}\]

   4. Taylor expanded around inf 0.8

      \[\leadsto \mathsf{fma}\left(x, \frac{1}{12}, \color{blue}{\left(\frac{1}{x} + \frac{1}{2}\right)}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0005386962406274384:\\ \;\;\;\;\sqrt[3]{\frac{e^{x}}{\mathsf{expm1}\left(x\right)} \cdot \left(\frac{e^{x}}{\mathsf{expm1}\left(x\right)} \cdot \frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{12}, \left(\frac{1}{2} + \frac{1}{x}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"

  :herbie-target
  (/ 1 (- 1 (exp (- x))))

  (/ (exp x) (- (exp x) 1)))