expax (section 3.5)

Average Error: 28.9 → 8.9

Time: 10.5s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -1.21980114088280676 \cdot 10^{-14}:\\ \;\;\;\;\frac{e^{2 \cdot \left(x \cdot a\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}\\ \mathbf{elif}\;a \cdot x \le 3.0774139795037859 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left({a}^{3}, \frac{1}{6} \cdot {x}^{3}, x \cdot \left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\left({\left(\sqrt[3]{e^{a \cdot x}}\right)}^{3} - 1\right) + \left(1 - 1\right)}} \cdot \sqrt[3]{\sqrt[3]{\left({\left(\sqrt[3]{e^{a \cdot x}}\right)}^{3} - 1\right) + \left(1 - 1\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}}\right)\\ \end{array}\]

C

double f(double a, double x) {
        double r102264 = a;
        double r102265 = x;
        double r102266 = r102264 * r102265;
        double r102267 = exp(r102266);
        double r102268 = 1.0;
        double r102269 = r102267 - r102268;
        return r102269;
}

↓

double f(double a, double x) {
        double r102270 = a;
        double r102271 = x;
        double r102272 = r102270 * r102271;
        double r102273 = -1.2198011408828068e-14;
        bool r102274 = r102272 <= r102273;
        double r102275 = 2.0;
        double r102276 = r102271 * r102270;
        double r102277 = r102275 * r102276;
        double r102278 = exp(r102277);
        double r102279 = 1.0;
        double r102280 = r102279 * r102279;
        double r102281 = r102278 - r102280;
        double r102282 = exp(r102272);
        double r102283 = r102282 + r102279;
        double r102284 = r102281 / r102283;
        double r102285 = 3.077413979503786e-19;
        bool r102286 = r102272 <= r102285;
        double r102287 = 3.0;
        double r102288 = pow(r102270, r102287);
        double r102289 = 0.16666666666666666;
        double r102290 = pow(r102271, r102287);
        double r102291 = r102289 * r102290;
        double r102292 = 0.5;
        double r102293 = pow(r102270, r102275);
        double r102294 = r102292 * r102293;
        double r102295 = r102294 * r102271;
        double r102296 = r102295 + r102270;
        double r102297 = r102271 * r102296;
        double r102298 = fma(r102288, r102291, r102297);
        double r102299 = r102282 - r102279;
        double r102300 = cbrt(r102299);
        double r102301 = r102300 * r102300;
        double r102302 = cbrt(r102282);
        double r102303 = pow(r102302, r102287);
        double r102304 = r102303 - r102279;
        double r102305 = r102279 - r102279;
        double r102306 = r102304 + r102305;
        double r102307 = cbrt(r102306);
        double r102308 = cbrt(r102307);
        double r102309 = r102308 * r102308;
        double r102310 = cbrt(r102300);
        double r102311 = r102309 * r102310;
        double r102312 = r102301 * r102311;
        double r102313 = r102286 ? r102298 : r102312;
        double r102314 = r102274 ? r102284 : r102313;
        return r102314;
}

Error

Bits error versus a

Bits error versus x

Target

Original	28.9
Target	0.2
Herbie	8.9

\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (* a x) < -1.2198011408828068e-14

   1. Initial program 1.0

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

   3. Applied flip-- 1.0

      \[\leadsto \color{blue}{\frac{e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1}{e^{a \cdot x} + 1}}\]

   4. Simplified 1.0

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

   if -1.2198011408828068e-14 < (* a x) < 3.077413979503786e-19

   1. Initial program 44.7

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

   2. Taylor expanded around 0 12.6

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

   3. Simplified 12.6

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

   if 3.077413979503786e-19 < (* a x)

   1. Initial program 28.0

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

   3. Applied add-cube-cbrt 28.1

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

   5. Applied add-cube-cbrt 28.1

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

   7. Applied add-cube-cbrt 28.1

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

   8. Applied add-cube-cbrt 28.9

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

   9. Applied prod-diff 29.0

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

   10. Simplified 29.0

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

   11. Simplified 29.0

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

   13. Applied add-cube-cbrt 29.0

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

   14. Applied add-cube-cbrt 29.2

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

   15. Applied prod-diff 29.2

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

   16. Simplified 29.3

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

   17. Simplified 29.3

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

4. Final simplification 8.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -1.21980114088280676 \cdot 10^{-14}:\\ \;\;\;\;\frac{e^{2 \cdot \left(x \cdot a\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}\\ \mathbf{elif}\;a \cdot x \le 3.0774139795037859 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left({a}^{3}, \frac{1}{6} \cdot {x}^{3}, x \cdot \left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\left({\left(\sqrt[3]{e^{a \cdot x}}\right)}^{3} - 1\right) + \left(1 - 1\right)}} \cdot \sqrt[3]{\sqrt[3]{\left({\left(\sqrt[3]{e^{a \cdot x}}\right)}^{3} - 1\right) + \left(1 - 1\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

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