expax (section 3.5)

Average Error: 29.4 → 10.2

Time: 4.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -6.2540867210978604 \cdot 10^{-60}:\\ \;\;\;\;\sqrt[3]{\frac{{\left(e^{\mathsf{fma}\left(a, x, a \cdot x\right)} - 1 \cdot 1\right)}^{3}}{{\left({\left(e^{a \cdot x}\right)}^{3} + {1}^{3}\right)}^{3}}} \cdot \mathsf{fma}\left(1, 1 - e^{a \cdot x}, e^{\mathsf{fma}\left(a, x, a \cdot x\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(0.16666666666666663, {a}^{3} \cdot {x}^{3}, 1 \cdot \left(a \cdot x\right)\right)\right)\\ \end{array}\]

C

double f(double a, double x) {
        double r66502 = a;
        double r66503 = x;
        double r66504 = r66502 * r66503;
        double r66505 = exp(r66504);
        double r66506 = 1.0;
        double r66507 = r66505 - r66506;
        return r66507;
}

↓

double f(double a, double x) {
        double r66508 = a;
        double r66509 = x;
        double r66510 = r66508 * r66509;
        double r66511 = -6.25408672109786e-60;
        bool r66512 = r66510 <= r66511;
        double r66513 = fma(r66508, r66509, r66510);
        double r66514 = exp(r66513);
        double r66515 = 1.0;
        double r66516 = r66515 * r66515;
        double r66517 = r66514 - r66516;
        double r66518 = 3.0;
        double r66519 = pow(r66517, r66518);
        double r66520 = exp(r66510);
        double r66521 = pow(r66520, r66518);
        double r66522 = pow(r66515, r66518);
        double r66523 = r66521 + r66522;
        double r66524 = pow(r66523, r66518);
        double r66525 = r66519 / r66524;
        double r66526 = cbrt(r66525);
        double r66527 = r66515 - r66520;
        double r66528 = fma(r66515, r66527, r66514);
        double r66529 = r66526 * r66528;
        double r66530 = 0.5;
        double r66531 = 2.0;
        double r66532 = pow(r66508, r66531);
        double r66533 = pow(r66509, r66531);
        double r66534 = r66532 * r66533;
        double r66535 = 0.16666666666666663;
        double r66536 = pow(r66508, r66518);
        double r66537 = pow(r66509, r66518);
        double r66538 = r66536 * r66537;
        double r66539 = r66515 * r66510;
        double r66540 = fma(r66535, r66538, r66539);
        double r66541 = fma(r66530, r66534, r66540);
        double r66542 = r66512 ? r66529 : r66541;
        return r66542;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.4
Target	0.2
Herbie	10.2

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

Derivation

1. Split input into 2 regimes

2. if (* a x) < -6.25408672109786e-60

   1. Initial program 7.9

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

   3. Applied add-cbrt-cube 7.9

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

   4. Simplified 7.9

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

   6. Applied flip-- 7.9

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

   7. Applied cube-div 7.9

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

   9. Applied prod-exp 7.8

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

   10. Simplified 7.8

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

   12. Applied flip3-+ 7.8

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

   13. Applied cube-div 7.8

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

   14. Applied associate-/r/ 7.8

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

   15. Applied cbrt-prod 7.8

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

   16. Simplified 7.8

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

   if -6.25408672109786e-60 < (* a x)

   1. Initial program 43.3

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

   3. Applied add-cbrt-cube 43.3

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

   4. Simplified 43.3

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

   6. Applied flip-- 43.3

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

   7. Applied cube-div 43.4

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

   8. Taylor expanded around 0 11.7

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(0.16666666666666663 \cdot \left({a}^{3} \cdot {x}^{3}\right) + 1 \cdot \left(a \cdot x\right)\right)}\]

   9. Simplified 11.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(0.16666666666666663, {a}^{3} \cdot {x}^{3}, 1 \cdot \left(a \cdot x\right)\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 10.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -6.2540867210978604 \cdot 10^{-60}:\\ \;\;\;\;\sqrt[3]{\frac{{\left(e^{\mathsf{fma}\left(a, x, a \cdot x\right)} - 1 \cdot 1\right)}^{3}}{{\left({\left(e^{a \cdot x}\right)}^{3} + {1}^{3}\right)}^{3}}} \cdot \mathsf{fma}\left(1, 1 - e^{a \cdot x}, e^{\mathsf{fma}\left(a, x, a \cdot x\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(0.16666666666666663, {a}^{3} \cdot {x}^{3}, 1 \cdot \left(a \cdot x\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 +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))