expq3 (problem 3.4.2)

Average Error: 60.3 → 54.8

Time: 10.9s

Precision: 64

• could not determine a ground truth for program body

  1. a = -2.2138166548721942e+229
  2. b = -9.77429183080671e+265
  3. eps = -1.6504517267772828e-18

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -1.949192232894134483079179646343452700776 \cdot 10^{49}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right) + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \varepsilon \cdot b\right)\right) \cdot \left(e^{a \cdot \varepsilon} - 1\right)}\\ \mathbf{elif}\;a \le 3.146231950313864865160087198758186099744 \cdot 10^{60}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\frac{1}{6} \cdot \left({a}^{3} \cdot {\varepsilon}^{3}\right) + \left(\frac{1}{2} \cdot \left({a}^{2} \cdot {\varepsilon}^{2}\right) + a \cdot \varepsilon\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{a \cdot \varepsilon} - 1\right)}\\ \end{array}\]

C

double f(double a, double b, double eps) {
        double r159409 = eps;
        double r159410 = a;
        double r159411 = b;
        double r159412 = r159410 + r159411;
        double r159413 = r159412 * r159409;
        double r159414 = exp(r159413);
        double r159415 = 1.0;
        double r159416 = r159414 - r159415;
        double r159417 = r159409 * r159416;
        double r159418 = r159410 * r159409;
        double r159419 = exp(r159418);
        double r159420 = r159419 - r159415;
        double r159421 = r159411 * r159409;
        double r159422 = exp(r159421);
        double r159423 = r159422 - r159415;
        double r159424 = r159420 * r159423;
        double r159425 = r159417 / r159424;
        return r159425;
}

↓

double f(double a, double b, double eps) {
        double r159426 = a;
        double r159427 = -1.9491922328941345e+49;
        bool r159428 = r159426 <= r159427;
        double r159429 = eps;
        double r159430 = b;
        double r159431 = r159426 + r159430;
        double r159432 = r159431 * r159429;
        double r159433 = exp(r159432);
        double r159434 = 1.0;
        double r159435 = r159433 - r159434;
        double r159436 = r159429 * r159435;
        double r159437 = 0.16666666666666666;
        double r159438 = 3.0;
        double r159439 = pow(r159429, r159438);
        double r159440 = pow(r159430, r159438);
        double r159441 = r159439 * r159440;
        double r159442 = r159437 * r159441;
        double r159443 = 0.5;
        double r159444 = 2.0;
        double r159445 = pow(r159429, r159444);
        double r159446 = pow(r159430, r159444);
        double r159447 = r159445 * r159446;
        double r159448 = r159443 * r159447;
        double r159449 = r159429 * r159430;
        double r159450 = r159448 + r159449;
        double r159451 = r159442 + r159450;
        double r159452 = r159426 * r159429;
        double r159453 = exp(r159452);
        double r159454 = r159453 - r159434;
        double r159455 = r159451 * r159454;
        double r159456 = r159436 / r159455;
        double r159457 = 3.146231950313865e+60;
        bool r159458 = r159426 <= r159457;
        double r159459 = pow(r159426, r159438);
        double r159460 = r159459 * r159439;
        double r159461 = r159437 * r159460;
        double r159462 = pow(r159426, r159444);
        double r159463 = r159462 * r159445;
        double r159464 = r159443 * r159463;
        double r159465 = r159464 + r159452;
        double r159466 = r159461 + r159465;
        double r159467 = r159430 * r159429;
        double r159468 = exp(r159467);
        double r159469 = r159468 - r159434;
        double r159470 = r159466 * r159469;
        double r159471 = r159436 / r159470;
        double r159472 = exp(r159449);
        double r159473 = r159472 - r159434;
        double r159474 = r159473 * r159454;
        double r159475 = r159436 / r159474;
        double r159476 = r159458 ? r159471 : r159475;
        double r159477 = r159428 ? r159456 : r159476;
        return r159477;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.3
Target	14.7
Herbie	54.8

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

Derivation

1. Split input into 3 regimes

2. if a < -1.9491922328941345e+49

   1. Initial program 54.7

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

   2. Taylor expanded around inf 54.7

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

   3. Taylor expanded around 0 48.5

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

   if -1.9491922328941345e+49 < a < 3.146231950313865e+60

   1. Initial program 63.7

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

   2. Taylor expanded around 0 56.9

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

   if 3.146231950313865e+60 < a

   1. Initial program 53.9

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

   2. Taylor expanded around inf 53.9

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

4. Final simplification 54.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.949192232894134483079179646343452700776 \cdot 10^{49}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right) + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \varepsilon \cdot b\right)\right) \cdot \left(e^{a \cdot \varepsilon} - 1\right)}\\ \mathbf{elif}\;a \le 3.146231950313864865160087198758186099744 \cdot 10^{60}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\frac{1}{6} \cdot \left({a}^{3} \cdot {\varepsilon}^{3}\right) + \left(\frac{1}{2} \cdot \left({a}^{2} \cdot {\varepsilon}^{2}\right) + a \cdot \varepsilon\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{a \cdot \varepsilon} - 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 
(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))))