expq3 (problem 3.4.2)

Average Error: 60.4 → 0.4

Time: 37.1s

Precision: 64

• could not determine a ground truth for program body

  1. a = 7.706155067127394e+81
  2. b = 6.88703791615021e+280
  3. eps = 8.102269269768392e-109

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)} = -\infty:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{elif}\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)} \le 1.570835326132957750197283763877702962205 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \end{array}\]

C

double f(double a, double b, double eps) {
        double r3582760 = eps;
        double r3582761 = a;
        double r3582762 = b;
        double r3582763 = r3582761 + r3582762;
        double r3582764 = r3582763 * r3582760;
        double r3582765 = exp(r3582764);
        double r3582766 = 1.0;
        double r3582767 = r3582765 - r3582766;
        double r3582768 = r3582760 * r3582767;
        double r3582769 = r3582761 * r3582760;
        double r3582770 = exp(r3582769);
        double r3582771 = r3582770 - r3582766;
        double r3582772 = r3582762 * r3582760;
        double r3582773 = exp(r3582772);
        double r3582774 = r3582773 - r3582766;
        double r3582775 = r3582771 * r3582774;
        double r3582776 = r3582768 / r3582775;
        return r3582776;
}

↓

double f(double a, double b, double eps) {
        double r3582777 = a;
        double r3582778 = b;
        double r3582779 = r3582777 + r3582778;
        double r3582780 = eps;
        double r3582781 = r3582779 * r3582780;
        double r3582782 = exp(r3582781);
        double r3582783 = 1.0;
        double r3582784 = r3582782 - r3582783;
        double r3582785 = r3582784 * r3582780;
        double r3582786 = r3582780 * r3582778;
        double r3582787 = exp(r3582786);
        double r3582788 = r3582787 - r3582783;
        double r3582789 = r3582780 * r3582777;
        double r3582790 = exp(r3582789);
        double r3582791 = r3582790 - r3582783;
        double r3582792 = r3582788 * r3582791;
        double r3582793 = r3582785 / r3582792;
        double r3582794 = -inf.0;
        bool r3582795 = r3582793 <= r3582794;
        double r3582796 = 1.0;
        double r3582797 = r3582796 / r3582777;
        double r3582798 = r3582796 / r3582778;
        double r3582799 = r3582797 + r3582798;
        double r3582800 = 1.5708353261329578e-35;
        bool r3582801 = r3582793 <= r3582800;
        double r3582802 = r3582801 ? r3582793 : r3582799;
        double r3582803 = r3582795 ? r3582799 : r3582802;
        return r3582803;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.4
Target	14.6
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))) < -inf.0 or 1.5708353261329578e-35 < (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0)))

   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 58.1

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

   3. Simplified 57.4

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

   4. Taylor expanded around 0 0.3

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

   if -inf.0 < (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))) < 1.5708353261329578e-35

   1. Initial program 3.1

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)} = -\infty:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{elif}\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)} \le 1.570835326132957750197283763877702962205 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :pre (and (< -1.0 eps) (< eps 1.0))

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

  (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))