expq3 (problem 3.4.2)

Average Error: 58.4 → 51.3

Time: 2.5m

Precision: 64

• could not determine a ground truth for program body

  1. a = 2.6476780725355068e+101
  2. b = -6.203135907666058e-146
  3. eps = 3.346115000419954e-91

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

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}\;b \le -3.2697307538617256 \cdot 10^{+104}:\\ \;\;\;\;\frac{\sqrt[3]{\left(e^{\varepsilon \cdot \left(b + a\right)} - 1\right) \cdot \left(\left(e^{\varepsilon \cdot \left(b + a\right)} - 1\right) \cdot \left(e^{\varepsilon \cdot \left(b + a\right)} - 1\right)\right)} \cdot \varepsilon}{\left(\left(\varepsilon \cdot a + \varepsilon \cdot \left(\left(\frac{1}{6} \cdot a\right) \cdot \left(\left(\varepsilon \cdot a\right) \cdot \left(\varepsilon \cdot a\right)\right)\right)\right) + \frac{1}{2} \cdot \left(\left(\varepsilon \cdot a\right) \cdot \left(\varepsilon \cdot a\right)\right)\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)}\\ \mathbf{elif}\;b \le 1.981037746615871 \cdot 10^{+71}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(b + a\right)} - 1\right)}{\left(\left(\frac{1}{6} \cdot \left(\varepsilon \cdot \left(\left(\left(\varepsilon \cdot b\right) \cdot \left(\varepsilon \cdot b\right)\right) \cdot b\right)\right) + \left(\left(\varepsilon \cdot b\right) \cdot \left(\varepsilon \cdot b\right)\right) \cdot \frac{1}{2}\right) + \varepsilon \cdot b\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(b + a\right)} - 1\right)}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(\sqrt[3]{\left(\varepsilon \cdot a + \varepsilon \cdot \left(\left(\frac{1}{6} \cdot a\right) \cdot \left(\left(\varepsilon \cdot a\right) \cdot \left(\varepsilon \cdot a\right)\right)\right)\right) + \frac{1}{2} \cdot \left(\left(\varepsilon \cdot a\right) \cdot \left(\varepsilon \cdot a\right)\right)} \cdot \left(\sqrt[3]{\left(\varepsilon \cdot a + \varepsilon \cdot \left(\left(\frac{1}{6} \cdot a\right) \cdot \left(\left(\varepsilon \cdot a\right) \cdot \left(\varepsilon \cdot a\right)\right)\right)\right) + \frac{1}{2} \cdot \left(\left(\varepsilon \cdot a\right) \cdot \left(\varepsilon \cdot a\right)\right)} \cdot \left(\left(\sqrt[3]{\varepsilon \cdot a} + \frac{1}{6} \cdot \left(\left(\varepsilon \cdot a\right) \cdot \sqrt[3]{\varepsilon \cdot a}\right)\right) + \frac{1}{36} \cdot \left(\sqrt[3]{\varepsilon \cdot a} \cdot \left(\left(\varepsilon \cdot a\right) \cdot \left(\varepsilon \cdot a\right)\right)\right)\right)\right)\right)}\\ \end{array}\]

C

double f(double a, double b, double eps) {
        double r22839563 = eps;
        double r22839564 = a;
        double r22839565 = b;
        double r22839566 = r22839564 + r22839565;
        double r22839567 = r22839566 * r22839563;
        double r22839568 = exp(r22839567);
        double r22839569 = 1.0;
        double r22839570 = r22839568 - r22839569;
        double r22839571 = r22839563 * r22839570;
        double r22839572 = r22839564 * r22839563;
        double r22839573 = exp(r22839572);
        double r22839574 = r22839573 - r22839569;
        double r22839575 = r22839565 * r22839563;
        double r22839576 = exp(r22839575);
        double r22839577 = r22839576 - r22839569;
        double r22839578 = r22839574 * r22839577;
        double r22839579 = r22839571 / r22839578;
        return r22839579;
}

↓

double f(double a, double b, double eps) {
        double r22839580 = b;
        double r22839581 = -3.2697307538617256e+104;
        bool r22839582 = r22839580 <= r22839581;
        double r22839583 = eps;
        double r22839584 = a;
        double r22839585 = r22839580 + r22839584;
        double r22839586 = r22839583 * r22839585;
        double r22839587 = exp(r22839586);
        double r22839588 = 1.0;
        double r22839589 = r22839587 - r22839588;
        double r22839590 = r22839589 * r22839589;
        double r22839591 = r22839589 * r22839590;
        double r22839592 = cbrt(r22839591);
        double r22839593 = r22839592 * r22839583;
        double r22839594 = r22839583 * r22839584;
        double r22839595 = 0.16666666666666666;
        double r22839596 = r22839595 * r22839584;
        double r22839597 = r22839594 * r22839594;
        double r22839598 = r22839596 * r22839597;
        double r22839599 = r22839583 * r22839598;
        double r22839600 = r22839594 + r22839599;
        double r22839601 = 0.5;
        double r22839602 = r22839601 * r22839597;
        double r22839603 = r22839600 + r22839602;
        double r22839604 = r22839583 * r22839580;
        double r22839605 = exp(r22839604);
        double r22839606 = r22839605 - r22839588;
        double r22839607 = r22839603 * r22839606;
        double r22839608 = r22839593 / r22839607;
        double r22839609 = 1.981037746615871e+71;
        bool r22839610 = r22839580 <= r22839609;
        double r22839611 = r22839583 * r22839589;
        double r22839612 = r22839604 * r22839604;
        double r22839613 = r22839612 * r22839580;
        double r22839614 = r22839583 * r22839613;
        double r22839615 = r22839595 * r22839614;
        double r22839616 = r22839612 * r22839601;
        double r22839617 = r22839615 + r22839616;
        double r22839618 = r22839617 + r22839604;
        double r22839619 = exp(r22839594);
        double r22839620 = r22839619 - r22839588;
        double r22839621 = r22839618 * r22839620;
        double r22839622 = r22839611 / r22839621;
        double r22839623 = cbrt(r22839603);
        double r22839624 = cbrt(r22839594);
        double r22839625 = r22839594 * r22839624;
        double r22839626 = r22839595 * r22839625;
        double r22839627 = r22839624 + r22839626;
        double r22839628 = 0.027777777777777776;
        double r22839629 = r22839624 * r22839597;
        double r22839630 = r22839628 * r22839629;
        double r22839631 = r22839627 + r22839630;
        double r22839632 = r22839623 * r22839631;
        double r22839633 = r22839623 * r22839632;
        double r22839634 = r22839606 * r22839633;
        double r22839635 = r22839611 / r22839634;
        double r22839636 = r22839610 ? r22839622 : r22839635;
        double r22839637 = r22839582 ? r22839608 : r22839636;
        return r22839637;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	58.4
Target	14.2
Herbie	51.3

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

Derivation

1. Split input into 3 regimes

2. if b < -3.2697307538617256e+104

   1. Initial program 51.2

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

   2. Taylor expanded around 0 46.0

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

   3. Simplified 42.1

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

   5. Applied add-cbrt-cube 42.1

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

   if -3.2697307538617256e+104 < b < 1.981037746615871e+71

   1. Initial program 61.4

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

   2. Taylor expanded around 0 55.0

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

   3. Simplified 55.0

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

   if 1.981037746615871e+71 < b

   1. Initial program 52.1

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

   2. Taylor expanded around 0 47.5

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

   3. Simplified 44.0

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

   5. Applied add-cube-cbrt 44.3

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

   6. Taylor expanded around 0 61.6

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\left(\color{blue}{\left(\frac{1}{36} \cdot \left({a}^{2} \cdot \left(e^{\frac{1}{3} \cdot \left(\log \varepsilon + \log a\right)} \cdot {\varepsilon}^{2}\right)\right) + \left(e^{\frac{1}{3} \cdot \left(\log \varepsilon + \log a\right)} + \frac{1}{6} \cdot \left(a \cdot \left(e^{\frac{1}{3} \cdot \left(\log \varepsilon + \log a\right)} \cdot \varepsilon\right)\right)\right)\right)} \cdot \sqrt[3]{\left(\varepsilon \cdot \left(\left(\frac{1}{6} \cdot a\right) \cdot \left(\left(a \cdot \varepsilon\right) \cdot \left(a \cdot \varepsilon\right)\right)\right) + a \cdot \varepsilon\right) + \frac{1}{2} \cdot \left(\left(a \cdot \varepsilon\right) \cdot \left(a \cdot \varepsilon\right)\right)}\right) \cdot \sqrt[3]{\left(\varepsilon \cdot \left(\left(\frac{1}{6} \cdot a\right) \cdot \left(\left(a \cdot \varepsilon\right) \cdot \left(a \cdot \varepsilon\right)\right)\right) + a \cdot \varepsilon\right) + \frac{1}{2} \cdot \left(\left(a \cdot \varepsilon\right) \cdot \left(a \cdot \varepsilon\right)\right)}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

   7. Simplified 44.1

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\left(\color{blue}{\left(\left(\left(\left(\varepsilon \cdot a\right) \cdot \left(\varepsilon \cdot a\right)\right) \cdot \sqrt[3]{\varepsilon \cdot a}\right) \cdot \frac{1}{36} + \left(\frac{1}{6} \cdot \left(\left(\varepsilon \cdot a\right) \cdot \sqrt[3]{\varepsilon \cdot a}\right) + \sqrt[3]{\varepsilon \cdot a}\right)\right)} \cdot \sqrt[3]{\left(\varepsilon \cdot \left(\left(\frac{1}{6} \cdot a\right) \cdot \left(\left(a \cdot \varepsilon\right) \cdot \left(a \cdot \varepsilon\right)\right)\right) + a \cdot \varepsilon\right) + \frac{1}{2} \cdot \left(\left(a \cdot \varepsilon\right) \cdot \left(a \cdot \varepsilon\right)\right)}\right) \cdot \sqrt[3]{\left(\varepsilon \cdot \left(\left(\frac{1}{6} \cdot a\right) \cdot \left(\left(a \cdot \varepsilon\right) \cdot \left(a \cdot \varepsilon\right)\right)\right) + a \cdot \varepsilon\right) + \frac{1}{2} \cdot \left(\left(a \cdot \varepsilon\right) \cdot \left(a \cdot \varepsilon\right)\right)}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 51.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.2697307538617256 \cdot 10^{+104}:\\ \;\;\;\;\frac{\sqrt[3]{\left(e^{\varepsilon \cdot \left(b + a\right)} - 1\right) \cdot \left(\left(e^{\varepsilon \cdot \left(b + a\right)} - 1\right) \cdot \left(e^{\varepsilon \cdot \left(b + a\right)} - 1\right)\right)} \cdot \varepsilon}{\left(\left(\varepsilon \cdot a + \varepsilon \cdot \left(\left(\frac{1}{6} \cdot a\right) \cdot \left(\left(\varepsilon \cdot a\right) \cdot \left(\varepsilon \cdot a\right)\right)\right)\right) + \frac{1}{2} \cdot \left(\left(\varepsilon \cdot a\right) \cdot \left(\varepsilon \cdot a\right)\right)\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)}\\ \mathbf{elif}\;b \le 1.981037746615871 \cdot 10^{+71}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(b + a\right)} - 1\right)}{\left(\left(\frac{1}{6} \cdot \left(\varepsilon \cdot \left(\left(\left(\varepsilon \cdot b\right) \cdot \left(\varepsilon \cdot b\right)\right) \cdot b\right)\right) + \left(\left(\varepsilon \cdot b\right) \cdot \left(\varepsilon \cdot b\right)\right) \cdot \frac{1}{2}\right) + \varepsilon \cdot b\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(b + a\right)} - 1\right)}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(\sqrt[3]{\left(\varepsilon \cdot a + \varepsilon \cdot \left(\left(\frac{1}{6} \cdot a\right) \cdot \left(\left(\varepsilon \cdot a\right) \cdot \left(\varepsilon \cdot a\right)\right)\right)\right) + \frac{1}{2} \cdot \left(\left(\varepsilon \cdot a\right) \cdot \left(\varepsilon \cdot a\right)\right)} \cdot \left(\sqrt[3]{\left(\varepsilon \cdot a + \varepsilon \cdot \left(\left(\frac{1}{6} \cdot a\right) \cdot \left(\left(\varepsilon \cdot a\right) \cdot \left(\varepsilon \cdot a\right)\right)\right)\right) + \frac{1}{2} \cdot \left(\left(\varepsilon \cdot a\right) \cdot \left(\varepsilon \cdot a\right)\right)} \cdot \left(\left(\sqrt[3]{\varepsilon \cdot a} + \frac{1}{6} \cdot \left(\left(\varepsilon \cdot a\right) \cdot \sqrt[3]{\varepsilon \cdot a}\right)\right) + \frac{1}{36} \cdot \left(\sqrt[3]{\varepsilon \cdot a} \cdot \left(\left(\varepsilon \cdot a\right) \cdot \left(\varepsilon \cdot a\right)\right)\right)\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :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))))