expq3 (problem 3.4.2)

Average Error: 60.4 → 53.4

Time: 16.9s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

• could not determine a ground truth for program body

  1. a = 1.2798302040214843e-73
  2. b = 3.3915171398033327e+295
  3. eps = 2.148424432870658e-101

\[-1 \lt \varepsilon \land \varepsilon \lt 1\]

Math

\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;a \le -2.3194785649155656 \cdot 10^{26}:\\ \;\;\;\;\frac{\varepsilon \cdot \sqrt[3]{{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}^{3}}}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, \left({\varepsilon}^{3} \cdot \left(b \cdot b\right)\right) \cdot b, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, b \cdot \varepsilon\right)\right)}\\ \mathbf{elif}\;a \le 6.5664357011814966 \cdot 10^{46}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {\varepsilon}^{3}, \mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {\varepsilon}^{2}, a \cdot \varepsilon\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, \left({\varepsilon}^{3} \cdot \left(b \cdot b\right)\right) \cdot b, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, b \cdot \varepsilon\right)\right)}\\ \end{array}\]

C

double f(double a, double b, double eps) {
        double r165445 = eps;
        double r165446 = a;
        double r165447 = b;
        double r165448 = r165446 + r165447;
        double r165449 = r165448 * r165445;
        double r165450 = exp(r165449);
        double r165451 = 1.0;
        double r165452 = r165450 - r165451;
        double r165453 = r165445 * r165452;
        double r165454 = r165446 * r165445;
        double r165455 = exp(r165454);
        double r165456 = r165455 - r165451;
        double r165457 = r165447 * r165445;
        double r165458 = exp(r165457);
        double r165459 = r165458 - r165451;
        double r165460 = r165456 * r165459;
        double r165461 = r165453 / r165460;
        return r165461;
}

↓

double f(double a, double b, double eps) {
        double r165462 = a;
        double r165463 = -2.3194785649155656e+26;
        bool r165464 = r165462 <= r165463;
        double r165465 = eps;
        double r165466 = b;
        double r165467 = r165462 + r165466;
        double r165468 = r165467 * r165465;
        double r165469 = exp(r165468);
        double r165470 = 1.0;
        double r165471 = r165469 - r165470;
        double r165472 = 3.0;
        double r165473 = pow(r165471, r165472);
        double r165474 = cbrt(r165473);
        double r165475 = r165465 * r165474;
        double r165476 = r165462 * r165465;
        double r165477 = exp(r165476);
        double r165478 = r165477 - r165470;
        double r165479 = 0.16666666666666666;
        double r165480 = pow(r165465, r165472);
        double r165481 = r165466 * r165466;
        double r165482 = r165480 * r165481;
        double r165483 = r165482 * r165466;
        double r165484 = 0.5;
        double r165485 = 2.0;
        double r165486 = pow(r165465, r165485);
        double r165487 = pow(r165466, r165485);
        double r165488 = r165486 * r165487;
        double r165489 = r165466 * r165465;
        double r165490 = fma(r165484, r165488, r165489);
        double r165491 = fma(r165479, r165483, r165490);
        double r165492 = r165478 * r165491;
        double r165493 = r165475 / r165492;
        double r165494 = 6.566435701181497e+46;
        bool r165495 = r165462 <= r165494;
        double r165496 = r165465 * r165471;
        double r165497 = pow(r165462, r165472);
        double r165498 = r165497 * r165480;
        double r165499 = pow(r165462, r165485);
        double r165500 = r165499 * r165486;
        double r165501 = fma(r165484, r165500, r165476);
        double r165502 = fma(r165479, r165498, r165501);
        double r165503 = exp(r165489);
        double r165504 = r165503 - r165470;
        double r165505 = r165502 * r165504;
        double r165506 = r165496 / r165505;
        double r165507 = r165496 / r165492;
        double r165508 = r165495 ? r165506 : r165507;
        double r165509 = r165464 ? r165493 : r165508;
        return r165509;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.4
Target	15.1
Herbie	53.4

\[\frac{a + b}{a \cdot b}\]

Derivation

1. Split input into 3 regimes

2. if a < -2.3194785649155656e+26

   1. Initial program 54.8

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

   2. Taylor expanded around 0 49.0

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

   3. Simplified 49.0

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {\varepsilon}^{3} \cdot {b}^{3}, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, b \cdot \varepsilon\right)\right)}}\]
   4. Using strategy rm

   5. Applied unpow3 49.0

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, {\varepsilon}^{3} \cdot \color{blue}{\left(\left(b \cdot b\right) \cdot b\right)}, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, b \cdot \varepsilon\right)\right)}\]

   6. Applied associate-*r* 48.3

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{\left({\varepsilon}^{3} \cdot \left(b \cdot b\right)\right) \cdot b}, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, b \cdot \varepsilon\right)\right)}\]
   7. Using strategy rm

   8. Applied add-cbrt-cube 48.4

      \[\leadsto \frac{\varepsilon \cdot \color{blue}{\sqrt[3]{\left(\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)\right) \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}}}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, \left({\varepsilon}^{3} \cdot \left(b \cdot b\right)\right) \cdot b, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, b \cdot \varepsilon\right)\right)}\]

   9. Simplified 48.4

      \[\leadsto \frac{\varepsilon \cdot \sqrt[3]{\color{blue}{{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}^{3}}}}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, \left({\varepsilon}^{3} \cdot \left(b \cdot b\right)\right) \cdot b, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, b \cdot \varepsilon\right)\right)}\]

   if -2.3194785649155656e+26 < a < 6.566435701181497e+46

   1. Initial program 63.8

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

   2. Taylor expanded around 0 56.5

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

   3. Simplified 56.5

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {\varepsilon}^{3}, \mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {\varepsilon}^{2}, a \cdot \varepsilon\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

   if 6.566435701181497e+46 < a

   1. Initial program 54.6

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

   2. Taylor expanded around 0 49.2

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

   3. Simplified 49.2

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {\varepsilon}^{3} \cdot {b}^{3}, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, b \cdot \varepsilon\right)\right)}}\]
   4. Using strategy rm

   5. Applied unpow3 49.2

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, {\varepsilon}^{3} \cdot \color{blue}{\left(\left(b \cdot b\right) \cdot b\right)}, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, b \cdot \varepsilon\right)\right)}\]

   6. Applied associate-*r* 48.3

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{\left({\varepsilon}^{3} \cdot \left(b \cdot b\right)\right) \cdot b}, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, b \cdot \varepsilon\right)\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 53.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -2.3194785649155656 \cdot 10^{26}:\\ \;\;\;\;\frac{\varepsilon \cdot \sqrt[3]{{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}^{3}}}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, \left({\varepsilon}^{3} \cdot \left(b \cdot b\right)\right) \cdot b, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, b \cdot \varepsilon\right)\right)}\\ \mathbf{elif}\;a \le 6.5664357011814966 \cdot 10^{46}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {\varepsilon}^{3}, \mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {\varepsilon}^{2}, a \cdot \varepsilon\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, \left({\varepsilon}^{3} \cdot \left(b \cdot b\right)\right) \cdot b, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, b \cdot \varepsilon\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :precision binary64
  :pre (and (< -1 eps) (< eps 1))

  :herbie-target
  (/ (+ a b) (* a b))

  (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))))