expax (section 3.5)

Average Error: 28.7 → 9.7

Time: 6.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -5.305313852292152365229972038607186846093 \cdot 10^{-6}:\\ \;\;\;\;\sqrt[3]{\frac{{\left(\mathsf{fma}\left(1, e^{a \cdot x} + 1, e^{2 \cdot \left(a \cdot x\right)}\right) \cdot \left(e^{a \cdot x} - 1\right)\right)}^{3}}{\sqrt{{\left(\mathsf{fma}\left(1, e^{a \cdot x} + 1, e^{a \cdot x + a \cdot x}\right)\right)}^{3}} \cdot \sqrt{{\left(\mathsf{fma}\left(1, e^{a \cdot x} + 1, e^{a \cdot x + a \cdot x}\right)\right)}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(0.1666666666666665186369300499791279435158, {a}^{3} \cdot {x}^{3}, 1 \cdot \left(a \cdot x\right)\right)\right)\\ \end{array}\]

C

double f(double a, double x) {
        double r89681 = a;
        double r89682 = x;
        double r89683 = r89681 * r89682;
        double r89684 = exp(r89683);
        double r89685 = 1.0;
        double r89686 = r89684 - r89685;
        return r89686;
}

↓

double f(double a, double x) {
        double r89687 = a;
        double r89688 = x;
        double r89689 = r89687 * r89688;
        double r89690 = -5.305313852292152e-06;
        bool r89691 = r89689 <= r89690;
        double r89692 = 1.0;
        double r89693 = exp(r89689);
        double r89694 = r89693 + r89692;
        double r89695 = 2.0;
        double r89696 = r89695 * r89689;
        double r89697 = exp(r89696);
        double r89698 = fma(r89692, r89694, r89697);
        double r89699 = r89693 - r89692;
        double r89700 = r89698 * r89699;
        double r89701 = 3.0;
        double r89702 = pow(r89700, r89701);
        double r89703 = r89689 + r89689;
        double r89704 = exp(r89703);
        double r89705 = fma(r89692, r89694, r89704);
        double r89706 = pow(r89705, r89701);
        double r89707 = sqrt(r89706);
        double r89708 = r89707 * r89707;
        double r89709 = r89702 / r89708;
        double r89710 = cbrt(r89709);
        double r89711 = 0.5;
        double r89712 = pow(r89687, r89695);
        double r89713 = pow(r89688, r89695);
        double r89714 = r89712 * r89713;
        double r89715 = 0.16666666666666652;
        double r89716 = pow(r89687, r89701);
        double r89717 = pow(r89688, r89701);
        double r89718 = r89716 * r89717;
        double r89719 = r89692 * r89689;
        double r89720 = fma(r89715, r89718, r89719);
        double r89721 = fma(r89711, r89714, r89720);
        double r89722 = r89691 ? r89710 : r89721;
        return r89722;
}

Error

Bits error versus a

Bits error versus x

Target

Original	28.7
Target	0.2
Herbie	9.7

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

Derivation

1. Split input into 2 regimes

2. if (* a x) < -5.305313852292152e-06

   1. Initial program 0.1

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

   3. Applied add-cbrt-cube 0.1

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

   4. Simplified 0.1

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

   6. Applied flip3-- 0.1

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

   7. Applied cube-div 0.1

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

   8. Simplified 0.1

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

   10. Applied add-sqr-sqrt 0.1

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

   12. Applied difference-cubes 0.1

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

   13. Simplified 0.1

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

   if -5.305313852292152e-06 < (* a x)

   1. Initial program 44.2

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

   3. Applied add-cbrt-cube 44.2

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

   4. Simplified 44.2

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

   6. Applied flip3-- 44.2

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

   7. Applied cube-div 44.3

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

   8. Simplified 44.3

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

   10. Applied add-sqr-sqrt 44.3

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

   11. Taylor expanded around 0 14.9

       \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(0.1666666666666665186369300499791279435158 \cdot \left({a}^{3} \cdot {x}^{3}\right) + 1 \cdot \left(a \cdot x\right)\right)}\]

   12. Simplified 14.9

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(0.1666666666666665186369300499791279435158, {a}^{3} \cdot {x}^{3}, 1 \cdot \left(a \cdot x\right)\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 9.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -5.305313852292152365229972038607186846093 \cdot 10^{-6}:\\ \;\;\;\;\sqrt[3]{\frac{{\left(\mathsf{fma}\left(1, e^{a \cdot x} + 1, e^{2 \cdot \left(a \cdot x\right)}\right) \cdot \left(e^{a \cdot x} - 1\right)\right)}^{3}}{\sqrt{{\left(\mathsf{fma}\left(1, e^{a \cdot x} + 1, e^{a \cdot x + a \cdot x}\right)\right)}^{3}} \cdot \sqrt{{\left(\mathsf{fma}\left(1, e^{a \cdot x} + 1, e^{a \cdot x + a \cdot x}\right)\right)}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(0.1666666666666665186369300499791279435158, {a}^{3} \cdot {x}^{3}, 1 \cdot \left(a \cdot x\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019344 +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))