expq3 (problem 3.4.2)

Average Error: 58.5 → 0.4

Time: 45.3s

Precision: 64

• could not determine a ground truth for program body

  1. a = 4.064654073173689e-98
  2. b = 1.1284288191611718e+226
  3. eps = 6.485322633399421e-190

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)} = -\infty:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{elif}\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)} \le 3.800949607953052 \cdot 10^{-06}:\\ \;\;\;\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

C

double f(double a, double b, double eps) {
        double r11241711 = eps;
        double r11241712 = a;
        double r11241713 = b;
        double r11241714 = r11241712 + r11241713;
        double r11241715 = r11241714 * r11241711;
        double r11241716 = exp(r11241715);
        double r11241717 = 1.0;
        double r11241718 = r11241716 - r11241717;
        double r11241719 = r11241711 * r11241718;
        double r11241720 = r11241712 * r11241711;
        double r11241721 = exp(r11241720);
        double r11241722 = r11241721 - r11241717;
        double r11241723 = r11241713 * r11241711;
        double r11241724 = exp(r11241723);
        double r11241725 = r11241724 - r11241717;
        double r11241726 = r11241722 * r11241725;
        double r11241727 = r11241719 / r11241726;
        return r11241727;
}

↓

double f(double a, double b, double eps) {
        double r11241728 = a;
        double r11241729 = b;
        double r11241730 = r11241728 + r11241729;
        double r11241731 = eps;
        double r11241732 = r11241730 * r11241731;
        double r11241733 = exp(r11241732);
        double r11241734 = 1.0;
        double r11241735 = r11241733 - r11241734;
        double r11241736 = r11241735 * r11241731;
        double r11241737 = r11241731 * r11241729;
        double r11241738 = exp(r11241737);
        double r11241739 = r11241738 - r11241734;
        double r11241740 = r11241731 * r11241728;
        double r11241741 = exp(r11241740);
        double r11241742 = r11241741 - r11241734;
        double r11241743 = r11241739 * r11241742;
        double r11241744 = r11241736 / r11241743;
        double r11241745 = -inf.0;
        bool r11241746 = r11241744 <= r11241745;
        double r11241747 = r11241734 / r11241729;
        double r11241748 = r11241734 / r11241728;
        double r11241749 = r11241747 + r11241748;
        double r11241750 = 3.800949607953052e-06;
        bool r11241751 = r11241744 <= r11241750;
        double r11241752 = r11241751 ? r11241744 : r11241749;
        double r11241753 = r11241746 ? r11241749 : r11241752;
        return r11241753;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	58.5
Target	14.2
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))) < -inf.0 or 3.800949607953052e-06 < (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1)))

   1. Initial program 62.1

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

   2. Taylor expanded around 0 0.1

      \[\leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}}\]

   if -inf.0 < (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))) < 3.800949607953052e-06

   1. Initial program 3.7

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)} = -\infty:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{elif}\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)} \le 3.800949607953052 \cdot 10^{-06}:\\ \;\;\;\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 
(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))))