expax (section 3.5)

Average Error: 29.2 → 9.4

Time: 4.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -3.56979547989466417 \cdot 10^{-7}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{\left(\log \left(\sqrt{\sqrt{e^{e^{a \cdot x} - 1}}}\right) + \log \left(\sqrt{\sqrt{e^{e^{a \cdot x} - 1}}}\right)\right) + \log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right)}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}\\ \mathbf{elif}\;a \cdot x \le 2.97473326445349939 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{\log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right) + \log \left(\sqrt{e^{\mathsf{fma}\left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}, \sqrt[3]{e^{a \cdot x}}, -1\right)}}\right)}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}\\ \end{array}\]

C

double f(double a, double x) {
        double r90292 = a;
        double r90293 = x;
        double r90294 = r90292 * r90293;
        double r90295 = exp(r90294);
        double r90296 = 1.0;
        double r90297 = r90295 - r90296;
        return r90297;
}

↓

double f(double a, double x) {
        double r90298 = a;
        double r90299 = x;
        double r90300 = r90298 * r90299;
        double r90301 = -3.569795479894664e-07;
        bool r90302 = r90300 <= r90301;
        double r90303 = exp(r90300);
        double r90304 = 1.0;
        double r90305 = r90303 - r90304;
        double r90306 = cbrt(r90305);
        double r90307 = exp(r90305);
        double r90308 = sqrt(r90307);
        double r90309 = sqrt(r90308);
        double r90310 = log(r90309);
        double r90311 = r90310 + r90310;
        double r90312 = log(r90308);
        double r90313 = r90311 + r90312;
        double r90314 = cbrt(r90313);
        double r90315 = r90306 * r90314;
        double r90316 = r90315 * r90306;
        double r90317 = 2.9747332644534994e-19;
        bool r90318 = r90300 <= r90317;
        double r90319 = 0.5;
        double r90320 = 2.0;
        double r90321 = pow(r90298, r90320);
        double r90322 = pow(r90299, r90320);
        double r90323 = r90321 * r90322;
        double r90324 = 0.16666666666666666;
        double r90325 = 3.0;
        double r90326 = pow(r90298, r90325);
        double r90327 = pow(r90299, r90325);
        double r90328 = r90326 * r90327;
        double r90329 = fma(r90324, r90328, r90300);
        double r90330 = fma(r90319, r90323, r90329);
        double r90331 = cbrt(r90303);
        double r90332 = r90331 * r90331;
        double r90333 = -r90304;
        double r90334 = fma(r90332, r90331, r90333);
        double r90335 = exp(r90334);
        double r90336 = sqrt(r90335);
        double r90337 = log(r90336);
        double r90338 = r90312 + r90337;
        double r90339 = cbrt(r90338);
        double r90340 = r90306 * r90339;
        double r90341 = r90340 * r90306;
        double r90342 = r90318 ? r90330 : r90341;
        double r90343 = r90302 ? r90316 : r90342;
        return r90343;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.2
Target	0.2
Herbie	9.4

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

Derivation

1. Split input into 3 regimes

2. if (* a x) < -3.569795479894664e-07

   1. Initial program 0.2

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

   3. Applied add-cube-cbrt 0.2

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

   5. Applied add-log-exp 0.2

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

   6. Applied add-log-exp 0.2

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

   7. Applied diff-log 0.2

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

   8. Simplified 0.2

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

   10. Applied add-sqr-sqrt 0.2

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

   11. Applied log-prod 0.2

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

   13. Applied add-sqr-sqrt 0.2

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

   14. Applied sqrt-prod 0.2

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

   15. Applied log-prod 0.2

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

   if -3.569795479894664e-07 < (* a x) < 2.9747332644534994e-19

   1. Initial program 45.1

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

   2. Taylor expanded around 0 13.5

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

   3. Simplified 13.5

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

   if 2.9747332644534994e-19 < (* a x)

   1. Initial program 25.4

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

   3. Applied add-cube-cbrt 25.5

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

   5. Applied add-log-exp 25.5

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

   6. Applied add-log-exp 31.7

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

   7. Applied diff-log 31.8

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

   8. Simplified 31.8

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

   10. Applied add-sqr-sqrt 31.9

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

   11. Applied log-prod 31.9

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

   13. Applied add-cube-cbrt 32.0

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

   14. Applied fma-neg 32.0

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

4. Final simplification 9.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -3.56979547989466417 \cdot 10^{-7}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{\left(\log \left(\sqrt{\sqrt{e^{e^{a \cdot x} - 1}}}\right) + \log \left(\sqrt{\sqrt{e^{e^{a \cdot x} - 1}}}\right)\right) + \log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right)}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}\\ \mathbf{elif}\;a \cdot x \le 2.97473326445349939 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{\log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right) + \log \left(\sqrt{e^{\mathsf{fma}\left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}, \sqrt[3]{e^{a \cdot x}}, -1\right)}}\right)}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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))