expq3 (problem 3.4.2)

Average Error: 58.6 → 51.1

Time: 1.6m

Precision: 64

• could not determine a ground truth for program body

  1. a = 6.746333998582945e+56
  2. b = -5.866854151042948e+138
  3. eps = -1.4548935032872371e-119

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -3.0120510137650696 \cdot 10^{+125}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(\varepsilon \cdot b + \left(\frac{1}{2} \cdot \left(\left(\varepsilon \cdot b\right) \cdot \left(\varepsilon \cdot b\right)\right) + \left(\varepsilon \cdot \left(\left(\left(\varepsilon \cdot b\right) \cdot \left(\varepsilon \cdot b\right)\right) \cdot b\right)\right) \cdot \frac{1}{6}\right)\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)}\\ \mathbf{elif}\;a \le 218302460416680.75:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(\frac{1}{2} \cdot \left(\left(\varepsilon \cdot a\right) \cdot \left(\varepsilon \cdot a\right)\right) + \left(\left(\left(\left(\varepsilon \cdot a\right) \cdot \left(\varepsilon \cdot a\right)\right) \cdot \left(a \cdot \frac{1}{6}\right)\right) \cdot \varepsilon + \varepsilon \cdot a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(\varepsilon \cdot b + \left(\frac{1}{2} \cdot \left(\left(\varepsilon \cdot b\right) \cdot \left(\varepsilon \cdot b\right)\right) + \left(\varepsilon \cdot \left(\left(\left(\varepsilon \cdot b\right) \cdot \left(\varepsilon \cdot b\right)\right) \cdot b\right)\right) \cdot \frac{1}{6}\right)\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)}\\ \end{array}\]

C

double f(double a, double b, double eps) {
        double r14664703 = eps;
        double r14664704 = a;
        double r14664705 = b;
        double r14664706 = r14664704 + r14664705;
        double r14664707 = r14664706 * r14664703;
        double r14664708 = exp(r14664707);
        double r14664709 = 1.0;
        double r14664710 = r14664708 - r14664709;
        double r14664711 = r14664703 * r14664710;
        double r14664712 = r14664704 * r14664703;
        double r14664713 = exp(r14664712);
        double r14664714 = r14664713 - r14664709;
        double r14664715 = r14664705 * r14664703;
        double r14664716 = exp(r14664715);
        double r14664717 = r14664716 - r14664709;
        double r14664718 = r14664714 * r14664717;
        double r14664719 = r14664711 / r14664718;
        return r14664719;
}

↓

double f(double a, double b, double eps) {
        double r14664720 = a;
        double r14664721 = -3.0120510137650696e+125;
        bool r14664722 = r14664720 <= r14664721;
        double r14664723 = eps;
        double r14664724 = b;
        double r14664725 = r14664720 + r14664724;
        double r14664726 = r14664723 * r14664725;
        double r14664727 = exp(r14664726);
        double r14664728 = 1.0;
        double r14664729 = r14664727 - r14664728;
        double r14664730 = r14664723 * r14664729;
        double r14664731 = r14664723 * r14664724;
        double r14664732 = 0.5;
        double r14664733 = r14664731 * r14664731;
        double r14664734 = r14664732 * r14664733;
        double r14664735 = r14664733 * r14664724;
        double r14664736 = r14664723 * r14664735;
        double r14664737 = 0.16666666666666666;
        double r14664738 = r14664736 * r14664737;
        double r14664739 = r14664734 + r14664738;
        double r14664740 = r14664731 + r14664739;
        double r14664741 = r14664723 * r14664720;
        double r14664742 = exp(r14664741);
        double r14664743 = r14664742 - r14664728;
        double r14664744 = r14664740 * r14664743;
        double r14664745 = r14664730 / r14664744;
        double r14664746 = 218302460416680.75;
        bool r14664747 = r14664720 <= r14664746;
        double r14664748 = exp(r14664731);
        double r14664749 = r14664748 - r14664728;
        double r14664750 = r14664741 * r14664741;
        double r14664751 = r14664732 * r14664750;
        double r14664752 = r14664720 * r14664737;
        double r14664753 = r14664750 * r14664752;
        double r14664754 = r14664753 * r14664723;
        double r14664755 = r14664754 + r14664741;
        double r14664756 = r14664751 + r14664755;
        double r14664757 = r14664749 * r14664756;
        double r14664758 = r14664730 / r14664757;
        double r14664759 = r14664747 ? r14664758 : r14664745;
        double r14664760 = r14664722 ? r14664745 : r14664759;
        return r14664760;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	58.6
Target	14.3
Herbie	51.1

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

Derivation

1. Split input into 2 regimes

2. if a < -3.0120510137650696e+125 or 218302460416680.75 < a

   1. Initial program 53.0

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

   2. Taylor expanded around 0 46.8

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

   3. Simplified 43.8

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

   if -3.0120510137650696e+125 < a < 218302460416680.75

   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

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

   3. Simplified 54.8

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

4. Final simplification 51.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -3.0120510137650696 \cdot 10^{+125}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(\varepsilon \cdot b + \left(\frac{1}{2} \cdot \left(\left(\varepsilon \cdot b\right) \cdot \left(\varepsilon \cdot b\right)\right) + \left(\varepsilon \cdot \left(\left(\left(\varepsilon \cdot b\right) \cdot \left(\varepsilon \cdot b\right)\right) \cdot b\right)\right) \cdot \frac{1}{6}\right)\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)}\\ \mathbf{elif}\;a \le 218302460416680.75:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(\frac{1}{2} \cdot \left(\left(\varepsilon \cdot a\right) \cdot \left(\varepsilon \cdot a\right)\right) + \left(\left(\left(\left(\varepsilon \cdot a\right) \cdot \left(\varepsilon \cdot a\right)\right) \cdot \left(a \cdot \frac{1}{6}\right)\right) \cdot \varepsilon + \varepsilon \cdot a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(\varepsilon \cdot b + \left(\frac{1}{2} \cdot \left(\left(\varepsilon \cdot b\right) \cdot \left(\varepsilon \cdot b\right)\right) + \left(\varepsilon \cdot \left(\left(\left(\varepsilon \cdot b\right) \cdot \left(\varepsilon \cdot b\right)\right) \cdot b\right)\right) \cdot \frac{1}{6}\right)\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 
(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))))