expax (section 3.5)

Average Error: 29.2 → 9.4

Time: 4.7s

Precision: 64

Math

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

↓

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

C

double f(double a, double x) {
        double r88720 = a;
        double r88721 = x;
        double r88722 = r88720 * r88721;
        double r88723 = exp(r88722);
        double r88724 = 1.0;
        double r88725 = r88723 - r88724;
        return r88725;
}

↓

double f(double a, double x) {
        double r88726 = a;
        double r88727 = x;
        double r88728 = r88726 * r88727;
        double r88729 = -3.569795479894664e-07;
        bool r88730 = r88728 <= r88729;
        double r88731 = exp(r88728);
        double r88732 = 1.0;
        double r88733 = r88731 - r88732;
        double r88734 = cbrt(r88733);
        double r88735 = r88734 * r88734;
        double r88736 = exp(r88733);
        double r88737 = sqrt(r88736);
        double r88738 = log(r88737);
        double r88739 = sqrt(r88737);
        double r88740 = log(r88739);
        double r88741 = r88740 + r88740;
        double r88742 = r88738 + r88741;
        double r88743 = cbrt(r88742);
        double r88744 = r88735 * r88743;
        double r88745 = 2.9747332644534994e-19;
        bool r88746 = r88728 <= r88745;
        double r88747 = 0.5;
        double r88748 = 2.0;
        double r88749 = pow(r88726, r88748);
        double r88750 = pow(r88727, r88748);
        double r88751 = r88749 * r88750;
        double r88752 = 0.16666666666666666;
        double r88753 = 3.0;
        double r88754 = pow(r88726, r88753);
        double r88755 = pow(r88727, r88753);
        double r88756 = r88754 * r88755;
        double r88757 = fma(r88752, r88756, r88728);
        double r88758 = fma(r88747, r88751, r88757);
        double r88759 = cbrt(r88731);
        double r88760 = r88759 * r88759;
        double r88761 = -r88732;
        double r88762 = fma(r88760, r88759, r88761);
        double r88763 = exp(r88762);
        double r88764 = sqrt(r88763);
        double r88765 = log(r88764);
        double r88766 = r88765 + r88738;
        double r88767 = cbrt(r88766);
        double r88768 = r88735 * r88767;
        double r88769 = r88746 ? r88758 : r88768;
        double r88770 = r88730 ? r88744 : r88769;
        return r88770;
}

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

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

   6. Applied add-log-exp 0.2

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

   7. Applied diff-log 0.2

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

   8. Simplified 0.2

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

   10. Applied add-sqr-sqrt 0.2

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

   11. Applied log-prod 0.2

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

   13. Applied add-sqr-sqrt 0.2

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

   14. Applied sqrt-prod 0.2

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

   15. Applied log-prod 0.2

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

   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

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

   6. Applied add-log-exp 31.7

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

   7. Applied diff-log 31.8

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

   8. Simplified 31.8

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

   10. Applied add-sqr-sqrt 31.9

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

   11. Applied log-prod 31.9

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

   13. Applied add-cube-cbrt 32.0

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

   14. Applied fma-neg 32.0

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

4. Final simplification 9.4

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

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))