expax (section 3.5)

Average Error: 28.9 → 7.2

Time: 20.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -6.875657620123969931839940451833229073664 \cdot 10^{-9}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{\frac{\frac{e^{\left(a \cdot x\right) \cdot 4} - {1}^{4}}{\mathsf{fma}\left(1, 1, {\left(e^{2}\right)}^{\left(a \cdot x\right)}\right)}}{1 + e^{a \cdot x}}}\\ \mathbf{elif}\;a \cdot x \le 8.25458109042831203908792331233248718025 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left({x}^{2}, \mathsf{fma}\left(\frac{1}{6} \cdot {a}^{3}, x, \frac{1}{2} \cdot {a}^{2}\right), a \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \frac{\sqrt[3]{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}}{{\left(e^{a \cdot x} + 1\right)}^{\frac{1}{3}}}\right) \cdot \sqrt[3]{\frac{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}{1 + e^{a \cdot x}}}\\ \end{array}\]

C

double f(double a, double x) {
        double r68961 = a;
        double r68962 = x;
        double r68963 = r68961 * r68962;
        double r68964 = exp(r68963);
        double r68965 = 1.0;
        double r68966 = r68964 - r68965;
        return r68966;
}

↓

double f(double a, double x) {
        double r68967 = a;
        double r68968 = x;
        double r68969 = r68967 * r68968;
        double r68970 = -6.87565762012397e-09;
        bool r68971 = r68969 <= r68970;
        double r68972 = exp(r68969);
        double r68973 = 1.0;
        double r68974 = r68972 - r68973;
        double r68975 = cbrt(r68974);
        double r68976 = r68975 * r68975;
        double r68977 = 4.0;
        double r68978 = r68969 * r68977;
        double r68979 = exp(r68978);
        double r68980 = pow(r68973, r68977);
        double r68981 = r68979 - r68980;
        double r68982 = 2.0;
        double r68983 = exp(r68982);
        double r68984 = pow(r68983, r68969);
        double r68985 = fma(r68973, r68973, r68984);
        double r68986 = r68981 / r68985;
        double r68987 = r68973 + r68972;
        double r68988 = r68986 / r68987;
        double r68989 = cbrt(r68988);
        double r68990 = r68976 * r68989;
        double r68991 = 8.254581090428312e-23;
        bool r68992 = r68969 <= r68991;
        double r68993 = pow(r68968, r68982);
        double r68994 = 0.16666666666666666;
        double r68995 = 3.0;
        double r68996 = pow(r68967, r68995);
        double r68997 = r68994 * r68996;
        double r68998 = 0.5;
        double r68999 = pow(r68967, r68982);
        double r69000 = r68998 * r68999;
        double r69001 = fma(r68997, r68968, r69000);
        double r69002 = fma(r68993, r69001, r68969);
        double r69003 = r68982 * r68969;
        double r69004 = exp(r69003);
        double r69005 = r68973 * r68973;
        double r69006 = r69004 - r69005;
        double r69007 = cbrt(r69006);
        double r69008 = r68972 + r68973;
        double r69009 = 0.3333333333333333;
        double r69010 = pow(r69008, r69009);
        double r69011 = r69007 / r69010;
        double r69012 = r68975 * r69011;
        double r69013 = r69006 / r68987;
        double r69014 = cbrt(r69013);
        double r69015 = r69012 * r69014;
        double r69016 = r68992 ? r69002 : r69015;
        double r69017 = r68971 ? r68990 : r69016;
        return r69017;
}

Error

Bits error versus a

Bits error versus x

Target

Original	28.9
Target	0.2
Herbie	7.2

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

Derivation

1. Split input into 3 regimes

2. if (* a x) < -6.87565762012397e-09

   1. Initial program 0.3

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

   3. Applied add-cube-cbrt 0.3

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

   5. Applied flip-- 0.3

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

   6. Simplified 0.3

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

   7. Simplified 0.3

      \[\leadsto \left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{\frac{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}{\color{blue}{1 + e^{a \cdot x}}}}\]
   8. Using strategy rm

   9. Applied flip-- 0.3

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

   10. Simplified 0.3

       \[\leadsto \left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{\frac{\frac{\color{blue}{e^{\left(a \cdot x\right) \cdot 4} - {1}^{4}}}{e^{2 \cdot \left(a \cdot x\right)} + 1 \cdot 1}}{1 + e^{a \cdot x}}}\]

   11. Simplified 0.3

       \[\leadsto \left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{\frac{\frac{e^{\left(a \cdot x\right) \cdot 4} - {1}^{4}}{\color{blue}{\mathsf{fma}\left(1, 1, {\left(e^{2}\right)}^{\left(a \cdot x\right)}\right)}}}{1 + e^{a \cdot x}}}\]

   if -6.87565762012397e-09 < (* a x) < 8.254581090428312e-23

   1. Initial program 44.5

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

   2. Taylor expanded around 0 13.0

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

   3. Simplified 9.9

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

   if 8.254581090428312e-23 < (* a x)

   1. Initial program 31.6

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

   3. Applied add-cube-cbrt 31.6

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

   5. Applied flip-- 32.5

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

   6. Simplified 32.2

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

   7. Simplified 32.2

      \[\leadsto \left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{\frac{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}{\color{blue}{1 + e^{a \cdot x}}}}\]
   8. Using strategy rm

   9. Applied flip-- 32.3

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

   10. Applied cbrt-div 32.3

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

   11. Simplified 31.7

       \[\leadsto \left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \frac{\color{blue}{\sqrt[3]{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}}}{\sqrt[3]{e^{a \cdot x} + 1}}\right) \cdot \sqrt[3]{\frac{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}{1 + e^{a \cdot x}}}\]
   12. Using strategy rm

   13. Applied pow1/3 31.7

       \[\leadsto \left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \frac{\sqrt[3]{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}}{\color{blue}{{\left(e^{a \cdot x} + 1\right)}^{\frac{1}{3}}}}\right) \cdot \sqrt[3]{\frac{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}{1 + e^{a \cdot x}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 7.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -6.875657620123969931839940451833229073664 \cdot 10^{-9}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{\frac{\frac{e^{\left(a \cdot x\right) \cdot 4} - {1}^{4}}{\mathsf{fma}\left(1, 1, {\left(e^{2}\right)}^{\left(a \cdot x\right)}\right)}}{1 + e^{a \cdot x}}}\\ \mathbf{elif}\;a \cdot x \le 8.25458109042831203908792331233248718025 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left({x}^{2}, \mathsf{fma}\left(\frac{1}{6} \cdot {a}^{3}, x, \frac{1}{2} \cdot {a}^{2}\right), a \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \frac{\sqrt[3]{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}}{{\left(e^{a \cdot x} + 1\right)}^{\frac{1}{3}}}\right) \cdot \sqrt[3]{\frac{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}{1 + e^{a \cdot x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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))